Eur. Phys. J. AP 16, 231–238 (2001)
THE EUROPEAN PHYSICAL JOURNAL APPLIED PHYSICS c EDP Sciences 2001
Numerical simulation of the interaction of biological cells with an ice front during freezing M. Carin1,a and M. Jaeger2 1 2
Universit´e de Bretagne Sud, Centre de Recherche, BP 92116, 56321 Lorient, France IUSTIb , 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France Received: 13 February 2001 / Revised: 18 July 2001 / Accepted: 20 July 2001 Abstract. The goal of this study is a better understanding of the interaction between cells and a solidification front during a cryopreservation process. This technique of freezing is commonly used to conserve biological material for long periods at low temperatures. However the biophysical mechanisms of cell injuries during freezing are difficult to understand because a cell is a very sophisticated microstructure interacting with its environment. We have developed a finite element model to simulate the response of cells to an advancing solidification front. A special front-tracking technique is used to compute the motion of the cell membrane and the ice front during freezing. The model solves the conductive heat transfer equation and the diffusion equation of a solute on a domain containing three phases: one or more cells, the extra-cellular solution and the growing ice. This solid phase growing from a binary salt solution rejects the solute in the liquid phase and increases the solute gradient around the cell. This induces the shrinkage of the cell. The model is used to simulate the engulfment of one cell modelling a red blood cell by an advancing solidification front initially planar or not is computed. We compare the incorporation of a cell with that of a solid particle. PACS. 44.05.+e Analytical and numerical techniques – 87. Biological and medical physics – 81.10.-h Methods of crystal growth; physics of crystal growth
1 Introduction The control of the incorporation of second phase particles into a crystal growing from its melt is an important challenge for many applications in material processing. Depending on the operating parameters, a given particle can be engulfed in the solid phase, pushed continuously or pushed for a certain distance before the entrapment occurs. The particle pushing phenomena is generally favored by the presence of small particles and slow cooling rates. Theoretical models have been developed to explain this phenomenon and to predict the critical velocity at which the transition from pushing to engulfment takes place (see [1] for a review, [2,3]). However, none of the existing models is able to adapt to all aspects of the problem. For example, the role played in the engulfment process by a difference in thermophysical properties among the system components is still not well established [4]. The main difficulty for the development of an analytical solution is probably the calculation of the evolution of the interface shape as the solidification front approaches the particle. Most models assume a flat interface or use an ad hoc approximation to the interface geometry near the a b
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particle. Recently, Wilde and coworkers [5,6] have shown experimentally that models based on such an assumption cannot deal with particle/dendrite interactions, which is the case often encountered in practice. The cryogenic preservation of biological materials is another application in which the understanding of problems of this kind is very important [7]. However, the goal is a little different since the product of interest is not the solid phase but the particles themselves. The aim is to preserve the biological properties of the cells during a freezing-thawing cycle. Indeed, cryopreservation is an adequate process to conserve some biological materials for long periods. However, ensuring the survival of cells is a big challenge. Since the beginning of cryopreservation, in the late fifties, and the discovery of the cryoprotectant properties of some substances, some progress has been made. Optimal freezing protocols have been determined experimentally for a few types of cells (including blood cells, embryos). However, a complete understanding of the biophysical mechanisms of freezing injuries is still lacking. The main reason is that even the simplest cell is a highly sophisticated microstructure, which interacts actively with its environment. In the case of freezing, large variations of solute concentrations occur during solidification. The osmotic pressure inside and outside cells is equilibrated by the osmotic loss of intracellular water,
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which causes substantial shrinkage. Therefore, a suspension of cells behaves like a highly complex coupled system during freezing. The phase change process modifies parameters to which cells are very sensitive and, in turn, cells influence the kinetics of solidification. All attempts to model this system have split the problem, focusing either on the osmotic response of cells in a solution during freezing or on the determination of the critical velocity for engulfment. In the first approach, the cell is generally considered in an environment where the solute and temperature fields are uniform and evolve during freezing according to the phase diagram ([8–10]). However, it is well known that the rejection of the solute at the advancing solidification front produces a solute concentration gradient [11]. A cell placed in such an environment will undergo an uneven flux of water through its boundaries due to the nonuniform osmotic pressure. To predict accurately the osmotic response of cells during freezing, the numerical model has to take into account the spatial variations of the solute concentration and temperature fields. In the second approach, the theoretical models determine the critical velocity for engulfment, considering only passive particles (i.e. the osmotic response of cells, which induces their shrinkage is never considered) [7]. However the particle size is an important parameter of the particle pushing phenomena [12]. So it appears that the osmotic response of cells and the crystal growth process can not be treated separately for this problem. It is noteworthy that all biological aspects (ultrastructure of the cytoplasm and the membrane) have been omitted in this description. The interior ultrastructure of the cells makes relatively little difference to osmotic in and outflow. The origin of the repulsive forces between the solidification front and a cell may also differ from the classical particle case due to the molecular nature of the membrane. A numerical approach to this problem can be of great help. The usefulness of numerical methods has already been demonstrated in crystal growth through the application of phase field models [13]. These latter models could in principle be extended to analyze the entrapment of solid particles. However, in the case of cells, additional phenomena must be taken into account, which cannot readily be handled by models of this kind. In the present paper, we demonstrate an alternative approach. We have developed a 2D numerical model to study the cryopreservation problem under simplifying assumptions. This model and its underlying assumptions are presented briefly in the next section. The ability of the model to predict the osmotic response of a cell in an evolving solute concentration field has been demonstrated [14]. As a first result, we have proposed a plausible mechanism of cell migration toward regions of lower solute concentrations. The ability of the model to deal with the liquid-solid phase change of a binary solution is demonstrated in [15]. Therefore, the investigation of the interaction between a cell and an ice front can be considered and examples of computations are reported below.
Water + solute cooling
ice
Ice front
cell
Fig. 1. Schematic representation of a cellular suspension during freezing.
2 Numerical model 2.1 Assumptions of the model The physical system under consideration is a suspension of cells in a binary solution of salt. This system is cooled below the freezing temperature leading to the formation of an ice front moving across the domain (see Fig. 1). The underlying assumptions are: (1) 2D or axial symmetry; (2) the cells are modeled as simple vesicles, consisting in a liquid medium enclosed by a semi-permeable membrane; (3) the intracellular and extracellular media are binary solutions with parameters chosen to approximate aqueous NaCl solution; (4) the solute is completely non-permeating and is totally rejected by the solid phase; (5) no fluid motion is induced by membrane forces or density variations, no cell motion is induced by gravity; (6) the mechanical action of membranes is not considered in this primary study; (7) the ice, the extracellular and intracellular media have constant phase-dependent properties; (8) the hydraulic permeability of the membrane is temperature dependent. 2.2 Equations of heat transfer and solute diffusion In the present state of development, the model solves the conductive heat transfer equation (1) and the diffusion equation (2) of a solute on a domain containing three phases: one or more cells, the extra-cellular solution and the growing ice: ρCp
∂T = div(k gradT ), ∂t
∂C = div(D gradC), ∂t
(1)
(2)
where T , the temperature and C, the solute concentration are the unknowns of the problem. The density ρ, the specific heat Cp , the thermal conductivity k and the solute diffusion coefficient D are the physical parameters characterizing each phase.
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2.3.2 Interfacial conditions for the concentration and temperature problems
2.3 Interfacial conditions 2.3.1 Velocity of the interfaces The different phases are separated by interfaces of two kinds, namely cell membranes and the solidification front. The position of these interfaces at time t + ∆t is given by: x(t + ∆t) = x(t) + ∆tV(x(t), t),
(3)
where V(x(t), t) is the interface velocity at time t and position x(t). The motion of the interface and thus the expression of the velocity V(x(t), t) depend on the nature of the interface considered. The motion of the ice front is deduced from the Stefan condition, which expresses the energy balance along the solid-liquid interface: ρ− ∆HVice = ρ− ∆HVice n − + ! ∂T ∂T = k− − k+ n. ∂n ∂n
−
Vm = Vm n = −LRT (C − C )n.
where V is the velocity of the membrane or the solidification front and the exponent ± refers to the right and left sides of the interface. For the temperature problem, the temperature is assumed continuous at the interfaces: T = T + = T −.
(8)
The temperature is specified along the ice front according to the Gibbs-Thomson relation: (4)
Here n defines the normal vector to the solid/liquid interface directed toward the liquid phase and ∆H is the latent heat of phase change. The signs − and + refer respectively to the solid and liquid phase. Although the model is able to take into account phase dependent physical properties, equal density is assumed in the solid and liquid phase. Indeed a jump in the density during liquid-solid phase change would imply a flow of the liquid phase, which is ignored in the present model. The velocity of the membrane, which is governed by osmosis, is proportional to the difference in the solute concentration on the internal cell side of the membrane C − and on the ambient side C + : +
The interfacial conditions for the concentration problem take into account the rejection of the solute by both the ice front and the cell membrane. This leads to the condition: ± ∂C ± ± ± D (gradC) n = D = −C ± Vn, (7) ∂n
(5)
Here n defines the normal vector to the membrane directed toward the external medium and R is the universal gas constant. L is the hydraulic permeability of the membrane and is temperature dependent according to the following equation: E 1 1 − , (6) L = Lref exp − R T Tref where E represents an energy of activation and Lref the hydraulic permeability at a given reference temperature Tref . In a homogeneous external medium, equation (5) leads to the shrinking of cells for hypertonic environment (Vm < 0) and to the swelling of cells in the hypotonic case (Vm > 0). Therefore, the normal velocity is proportional to the thermal flux discontinuity for a solidification front and to the solute concentration discontinuity for a membrane, where the important physical parameters are the latent heat of fusion and the membrane permeability respectively.
T = T0 + mC + −
γT0 κ. ρ∆H
(9)
Here T0 represents the equilibrium melting temperature of the pure substance (water), m is the slope of the linearized liquidus line from an equilibrium phase diagram, C + represents the solute concentration of the liquid phase, γ is the surface tension and κ is the curvature of the interface. For the case of a salt solution, the slope m is negative, inducing the decreasing of the melting point with the increase of salt concentration. 2.4 Front-tracking method The numerical challenge relies on the moving interfaces. An accurate representation of the discontinuity conditions is crucial for this problem and precludes conventional approaches based on a diffuse representation of the interfaces. We have developed a front tracking method to compute discontinuous solutions on unstructured finite element meshes made of three-node triangular finite elements (T3). The method is based on the following principles: - a finite element mesh covering the computational domain is initially defined. This mesh is the “reference domain-mesh”; - the temperature and solute concentration fields are computed on an interface fitted domain-mesh. This “computational domain-mesh” is built as new at each time step by moving the nearest nodes of the “reference domain-mesh” to the updated locations of the interfaces; - to account for fields subject to jump conditions at the interface (e.g. solute concentration field), the previous mesh fitting procedure is complemented by a special interfacial jump treatment. Each node of the domainmesh that is located on an interface is duplicated. One of the two nodes is assigned to the elements located on
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one side of the interface and the other one to the elements located on the other side. Therefore, the jump condition is the default mode. Continuity must be enforced when desired. Moreover, specific boundary elements (“jump-elements”) are generated along each interface, connecting two subsequent couples of nodes. The function of jump-elements is to account for interfacial fluxes, but they also enable the communication between the elements located on each side of an interface; - each interface is represented by its own mesh (“interface-mesh”) made of two-node linear elements (L2). At each time step the location of each interface is updated by moving its nodes in a Lagrangian manner. The local curvature and normal vector are computed on the interfacial-meshes using a cubic spline interpolation; - the data needed to give an accurate description of the interface, e.g. location, curvature, normal vector, right and left values of the different physical fields, are stored as nodal properties of the interface-mesh nodes. A straightforward transfer of data between interfacialmesh and domain-mesh provides the means to accurately reconstruct the fields of interest on the computational domain-mesh before starting the computation of the temperature and solute concentration fields. The node motion involved in the interface mesh fitting procedure must be accounted for in the equation by adding a convection term. This is typical of an “Arbitrary Lagrangian Eulerian” (ALE) approach. The velocity involved is that of the nodes (with a minus sign). However, in order to be able to deal with discontinuous fields, we consider instead in our method a fictitious mesh motion different from the real node motion. Actually, the nodes moved to the interface are considered as already located on the interface on the previous time step. The velocity involved in the ALE formulation is therefore the interface velocity. We have therefore called this method the “Front Tracking ALE” method. 2.5 Numerical model and algorithm The numerical model combines the Front Tracking ALE method and a 2D or axisymmetric finite element method. The ALE formulation of the evolution equations (1) and (2) are ∂T ALE ρCp −V · gradT = div(k gradT ), (10) ∂t ∂C − V ALE · gradC = div(D gradC), ∂t
(11)
where VALE is the velocity associated to the fictitious mesh motion of the Front Tracking ALE method. The solute diffusion problem and the heat transfer problem are solved on a domain Ω, with boundary ∂Ω, containing the three phases: cells, extracellular solution
and ice. The integral forms associated with the weak formulations of these two problems are Z ∂T IT = − V ALE · gradT ρCp δT ∂t Ω + k gradδT · gradT dS Z Z ∂T dl − − kδT ρ− ∆HδT Vice dl (12) ∂n ∂Ω Γice Z ∂C ALE δC −V IC = · gradC ∂t Ω Z ∂C dl DδC + D gradδC · gradC dS − ∂n ∂Ω Z Z Z − δCC + Vice dl − δC + C + Vm dl + δC − C − Vm dl Γice
Γm
Γm
(13) where δT and δC are weighting functions. In (12), the boundary integral on the solidification front Γice cancels as the temperature is fixed there by the Gibbs-Thomson condition (9). In (13), the contribution of the three boundary integrals on the solidification front Γice and on the membrane Γm corresponds to the solute concentration fluxes induced by the motion of the interfaces, as given by (7). In both integral forms (12) and (13), the integrals associated with the boundaries ∂Ω are zero for Dirichlet or homogeneous Neumann conditions. We use a three-node triangular element (T3) with a piece-wise linear approximation for the solute concentration and the temperature. For the time discretization we use the first order backward difference scheme, which is implicit. The organization of the computations for each time step is as follows: Interface updates: - update of the interfaces locations; - update of the topological properties of the interfaces via a cubic spline interpolation; - update of the interfacial-mesh. Building of the interface fitted computational domainmesh: - building of the computational domain-mesh; - update of the material properties; - transfer of data from the interfacial mesh to the computational domain-mesh. Solving the continuum mechanic problems: - computation of the solute concentration field; - computation of the temperature field; - transfer of data to the interfacial-mesh. More details about this model as well as a comprehensive convergence study can be found in [15]. In Figure 2 a close-up of the computational mesh in the cell region illustrates the efficiency of the mesh-fitting algorithm. The bold curves represent the solidification front and the cell membrane. As can be seen, only the nodes nearest the interfaces are displaced to the interfaces.
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6
3
0 Fig. 3. Incorporation of cell in a growing ice front. Evolution of the position of the ice front and the cell membrane.
-6 0
5
10
Fig. 2. Computational mesh.
3 Examples of particle-front interaction
Concentration (mol/m3)
450
-3
0 -10
The forces that are responsible for the pushing of particles by the solidification front are short-range molecular forces. We have not yet considered these forces in our model. Therefore, we report hereafter examples of computations for cooling rates sufficiently large that these forces are never dominant in comparison to the long-range solute field interactions. These interactions may dominate over the short-range molecular forces for front speeds much greater than the critical velocity for entrapment. The obstruction of the diffusion field by a particle leads to a reduction of the local growth velocity of the ice front just beneath it. As a result, the ice grows around the particle, leaving a solute-rich band all around it and in a narrow channel behind [16]. The distance between the ice front and the particle remains large relatively to the length scale at which short-range molecular forces act. In the two following sections (3.1 and 3.2), we present computations that have been done in cylindrical coordinates under the assumption that the problem is axially symmetric. Section 3.3 deals with a 2D computation.
3.1 Engulfment of a spherical cell by a flat ice front The first example concerns the engulfment of a spherical cell by a flat ice front. The initial radius of the cell (4 × 10−6 m) and the membrane permeability (1.66 × 10−12 m/s Pa) are those of a red blood cell (idealized here as a sphere). The initial solute concentration is isotonic, uniform and equal to 154 mol/m3 . The cooling rate is equal to −0.3 K/s and is subscribed to the left boundary of a rectangular domain. A zero flux condition is imposed along the upper and lower boundaries as well as the right boundary. Thus the planar front advances toward the right side with a growing rate of around 20 × 10−6 m/s, which
Nondimensional length
10
Fig. 4. Solute concentration profile at different time steps (∆t = 0.05 s).
is about twenty times larger than the critical velocity determined experimentally by K¨orber [7]. The cell is located initially at a distance of 52.5 × 10−6 m from the ice front. The evolution of the location of the solidification front and the cell membrane is depicted in Figure 3. As the front advances, the cell shrinks. When the front reaches the cell, the volume loss is equal to 40% of the initial cell volume. It can been seen that the presence of the cell in the liquid phase causes the solid-liquid interface to become concave. Indeed, Figure 4 shows the solute concentration profile in a cross section through the middle of the cell (axis of symmetry) at different time steps (∆t = 0.05 s). The nondimensional length represents the length divided by the initial cell radius. As the solidification front advances, the solute is rejected in the ambient aqueous solution, leading to the increase of the solute concentration in the liquid phase. Therefore the value of the melting temperature is lowered as predicted by equation (9). However, because of the blocking action of the cell, the solute concentration increases more in the gap between the ice front and the cell. As a result, the front curves as it approaches the cell. The obstruction of diffusion by the cell is obvious in Figure 4. The drop in concentration across the cell membrane becomes larger with time since the ambient salt concentration increases too fast for the cell to equilibrate its intracellular concentration with the environment (characteristic time of order 2 s). The net migration of the cell down the concentration gradient is also very small (3% of the initial cell radius) for the same reason. It can also be noticed that the intracellular concentration is quite uniform due to the small size of the cell.
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solid particle cell
Fig. 5. Comparison of the cases of cell (black) and particle (grey) at the same time step for a cooling rate of −0.3 K/s.
The large jump in concentration across the cell membrane is consistent with the theory of Mazur [17]. This theory attributes the decrease of the cell survival at high cooling rates by the formation of ice in the cells. The cells would not have enough time to lose a sufficient quantity of water to avoid intracellular crystallization. This generally leads to the death of the cell. Our model illustrates the competition between the kinetics of freezing and the kinetics of water loss through the membrane. Moreover the model exhibits a crystal growth around the cell, which is quite similar to experimental observations published by Ishiguro and Rubinsky [16]. In their experiments, the ice grows around red blood cells leading to the formation of arms separated by a channel. This channel contains an enriched solute solution that prevents the coalescence of the arms after the engulfment of the cell. 3.2 Effect of the osmotic response of the cell The fact that the curvature of the solidification front is influenced by the osmotic response of the cell is demonstrated in Figure 5. In this figure, we compare the locations of the interfaces in the case of the interaction with a solid (impermeable) particle, represented by the grey curves with the locations of the interfaces corresponding to the cell case. Except for the permeability, the physical properties of the particle are set equal to those of the cell. The cell and the particle have initially the same size. Thus, during freezing, the cell becomes smaller than the particle inducing a different deformation of the solidification front. It appears that the ice front advances more slowly in the presence of the solid particle in the region of this obstacle. However the gap between the cell and the front is nearly the same as the gap between the particle and the front. This indicates that the cell water loss is too small to have a significant effect on the solute concentration field. The difference in front velocity is mainly due to the difference of particle size. A large particle will induce a high obstruction of diffusion and a smaller front velocity. Therefore the phenomena of the engulfment seems very similar in both cases, the solidification front being little influenced by the rejection of water by the cell in the extracellular medium. Indeed, at −0.3 K/s, the cell has not enough time to
Fig. 6. Comparison of two cooling rates: −0.1 K/s (black) and −0.3 K/s (grey).
equilibrate its inside concentration with the outside leading to a high jump of concentration through its membrane as shown previously in Figure 4. However the influence of the flux of water through the membrane could be more important at lower cooling rates since in that case the cell manages to equilibrate its intracellular concentration with the extracellular concentration. The study of the engulfment of a particle or a cell seems to show that the blockage of the solute by the obstacle is a dominant effect. This has been confirmed by changing the thermophysical properties of the particle. The results obtained for a particle of much larger thermal conductivity have shown a very similar deformation of the front. This is not the case of the solidification of a pure substance. Indeed in the presence of a particle less conducting than the liquid the front becomes convex on approaching the particle. It becomes concave in the case of a particle more conducting than the melt [18]. The evolution of the ice front during the engulfment process depends on the cooling rate. This is illustrated in Figure 6, where we have compared the location of the interfaces (ice front and cell membrane) for two computations at two different cooling rates. It can been noticed that the channel size increases with the cooling rate as the gap does between the cell and the solidification front. This is clearly related to the solute concentration, which increases more at higher cooling rates (or high front speeds) than at smaller ones. It explains why particles are pushed forward at low cooling rates and incorporated at high ones. Indeed, a small gap between the cell and the solidification front will lead to larger repulsive forces, which may thus keep the cell ahead of the interface for longer. On the other hand, higher cooling rates lead to a larger channel size, smaller translational forces on the cell and therefore the possibility of engulfment of the cell by the ice front. 3.3 Interaction of a cell with a destabilized ice front The last example concerns a 2D computation of the interaction of a cell (identical to the one considered in the first example) with a flat ice front which evolves in cellular crystal growth. This case is of great importance for
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6
3
0
-3
-6
0
5
10
15
20
25
Fig. 7. Evolution of the interfaces in a case of a cell interacting with a destabilized ice front.
the study of the freezing of cells. Indeed, several studies have suggested that the morphology of the ice crystals and the mode of interaction between the ice crystals and the cells are important parameters for the cell survival [16,19, 20]. During the freezing of a physiological saline solution, the ice crystals can have two morphologies: a flat interface obtained for low cooling rates and a cellular interface occurring with higher cooling rates. To obtain a cellular growth in the computation, we have initially destabilized a planar front with a sinusoidal perturbation of amplitude equal to 3% of the wavelength. The evolution of the interfaces for a cooling rate of −0.5 K/s is shown in Figure 7. The cell is initially located at a distance of 55 × 10−6 m from the ice front ahead of the middle cellular dendrite. The computation shows that the amplitude of the perturbation of the front shape grows as long as the front is far from the cell, which is consistent with the theory of instability in crystal growth [21]. However, on approaching the cell, the bump in the center of the front begins to decrease and disappears during the engulfment of the cell. The channel formed around the cell is then larger than for a planar front. Therefore the osmotic response of the cell will be different according to the type of crystal growth. In the case of a cellular growth, the large channel will induce a higher shrinkage of the cell before the cell is completely surrounded by the ice. This could have a consequence on the cell survival since the cell will be exposed to very high solute concentrations during a long period, which is toxic for the cell. This demonstrates the importance of taking into account the interaction between the particles and the phase change interface when analyzing problems of this kind.
4 Conclusion Models of cryopreservation can be of great help for the understanding of the mechanisms involved in cryobiology, but the development of programs able to simulate accurately a cryopreservation procedure is still a difficult task. We have surmounted this difficulty by developing a computer code able to take into account the coupling of heat and mass transfers associated with the presence of moving interfaces which present discontinuities of material properties and variables. The dynamic behavior of a solidification
front in a presence of a cell or a particle can then be predicted. Unlike those used previously, this model can simulate the osmotic response of a cell exposed to a nonuniform temperature and solute concentration fields created by the propagation of a solid-liquid interface. Therefore the behavior of cells during freezing can be studied in situations more realistic than those considered in the previous models. The different computations have shown that a strong interaction exists between a cell and an advancing solidification front. The presence of the cell induces a local perturbation of the solute concentration field leading to a slowing of the ice front and to its growth around the cell. The osmotic response of the cell is then influenced by the deformation of the front and the size of the resulting channel. The knowledge of the change of the cell environment during the engulfment is particularly important because the cell is exposed to very high concentrations that affect the cell survival. Therefore the different quantities calculated by the model can be used to predict cryoinjury due to the cell shrinkage or the chemical toxicity and thus are potentially very important to cryobiologists. The comparison between the engulfment of a cell and a solid particle by a solidification front has shown that the evolution of the interface approaching the obstacle is very similar when the front velocity is much higher than the critical velocity below which the particles are pushed by the interface. The gap between the cell and the front is comparable to the gap between the particle and the front. Little influence of the thermophysical properties of the particle is observed. These simulations demonstrate that the diffusion of the solute is a dominant effect for the study of the interaction between a cell or a particle and a solidification front in this range of cooling rates. The study of the influence of the cooling rate on the engulfment of cell has shown that the morphology of the ice front is greatly affected by the cooling rate. At a lower cooling rate, the gap between the front and the cell is reduced as well as the curvature of the interface. This has been explained by the competition between the diffusion of the solute that is rejected ahead of the front and the velocity of the interface. Thus by reducing the cooling rate down to the critical velocity characterizing the particle repulsion phenomena, the gap between the cell and the
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solid-liquid interface will be sufficiently small so that the repulsion forces can appear. The last computation has illustrated the influence of the presence of cell on the morphology of the interface. In the case of a destabilized solidification front, the presence of cells blocks the growth of the bump ahead of the cell, the resulting channel is then larger, and so affects the osmotic response of the cell. Therefore with a better understanding of the interaction between the cells and the ice front, we could improve the cell survival by favoring the pattern of crystal growth that is less injurious. The simulations have shown how the particle engulfment can be favored when the interface velocity is higher than the critical velocity. High cooling rates induce the formation of a large channel. So very low cooling rates will lead to a smaller gap between the particle and the solidification front and the repulsion forces will appear. Future developments of the model will focus on the introduction of the short range molecular forces that are responsible for the repulsion of the particles. The particle pushing phenomena could then be simulated and analyzed at very low cooling rates. However, despite the simplifying assumptions, the model we have developed has given preliminary results that are consistent with the experimental observations. Funding for this work was provided by the French Minist`ere de la D´efense through DGA Contract n◦ 993404400/DSP. We wish to thank J. Wolfe for helpful discussions concerning the non-military aspects.
Nomenclature T C t x V n ρ Cp k D ∆H
temperature solute concentration time position velocity of the interface normal vector density specific heat thermal conductivity solute diffusion coefficient latent heat of phase change
L R T0 m γ κ Ω ∂Ω Γ
hydraulic permeability universal gas constant equilibrium melting temperature slope of the phase diagram surface tension curvature of the interface domain boundary of the domain Ω interface
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