Comparison between Ideal and Nonideal Solution Models for Single

J. Phys. Chem. B 2009, 113, 4853–4864

4853

Comparison between Ideal and Nonideal Solution Models for Single-Cell Cryopreservation Protocols Jaime Saenz,† Mehmet Toner,‡ and Ramon Risco*,† Escuela Superior de Ingenieros, UniVersidad de SeVilla, 41092 SeVilla, Spain, The Center for Engineering in Medicine, Massachusetts General Hospital, HarVard Medical School, 51 Blossom Street, Boston, Massachusetts 02114 ReceiVed: August 14, 2008; ReVised Manuscript ReceiVed: December 3, 2008

Models for cell dehydration during a cryopreservation protocol are usually based on the hypothesis of ideal dilute solution. The strong electrolyte character of NaCl makes us revisit these models. The case of nonideal solution is analyzed by computing the dehydration curves without this additional hypothesis. The conclusion is that, in general, while the application of the ideal dilute solution hypothesis is convenient in many cases, for some specific cooling rates there exist important differences in the degree of dehydration predicted by these two models in the studied cases of mouse sperm and hepatocyte. It is shown how this finding has relevant implications for the design and optimization of cryopreservation protocols. 1. Introduction Two cornerstones that much of cryobiology relies on are the discovery of cryoprotectants1,2 and the thermodynamic modeling of the cryopreservation processes.3 In this paper we shall tackle some aspects of this second issue in the case of single cells. The seminal work of Mazur3 was able to explain the inverted U-shaped behavior of cell survival as a function of the cooling rate, giving rise to what is nowadays called “the two-factors hypothesis”. This approach can be used as an important tool for the design of an optimal cryopreservation protocol for single cells: from the knowledge of the membrane water permeability at a given temperature, the cell surface area, and volume, it is possible to figure out the degree of supercooling of the intracellular solution, and from it the likelihood of intracellular ice formation. An exhaustive list of the biophysical hypotheses which this model relies on is displayed at the end of the work.3 They are as follows: (1) The plasma membrane remains intact during cooling, (2) the plasma membrane is permeable only to water, (3) the protoplasm behaves as an ideal dilute solution, (4) the external solution is always in equilibrium with ice, (5) the molar volume of water is substituted for its partial molar volume (dilute solution), (6) the cooling rate is constant, and (7) only the cell volume and the temperature are considered variables. We have focused our attention on one of these hypotheses, that is, to consider the intracellular solution as an ideal dilute solution. By definition, an ideal dilute solution is a solution in which the molecules of solute only interact with those of the solvent. This is especially valid for solutions of nonelectrolyte species. If strong electrolytes are present in the solution, such as NaCl, the electromagnetic forces between the ions of the solute give rise to important interactions among them and, therefore, even under very high dilution (we shall keep the hypothesis of dilute solution throughout the paper), the model of ideal solution is often no longer valid. We shall deal with this subtle aspect not only to get a deep insight into the * To whom correspondence should be addressed. E-mail: [email protected]. † Universidad de Sevilla. ‡ Harvard Medical School.

biophysical aspects of the cryopreservation but also to meet the goal of contributing to a rational approach to the optimization of the big number of parameters present in the cryopreservation process. This can allow improvements in the viability of interesting cryopreserved cells like sperm, oocytes, or umbilical cord blood or allow scaling to bigger biological samples. However, most of the efforts in the literature have been focused on the tedious experimental optimizations of these procedures.4 The question of the nonideality of the intracellular solution has been tackled a few times by different authors.5-9 Being a difficult issue,5 only partial aspects have been addressed in some highly valuable works. Levin et al.6 focused his attention on erythrocytes; in this case the nonideality of the intracellular solution for a linear cooling profile in the regime of high cooling rates (-5000 °C/min) was explored. The nonideal behavior of other hemic components was also studied by Arnoud and Pegg7 and by Hunt et al.8 Outside the realm of cryopreservation, the role of the nonideality in the renal concentrating mechanism was also studied by Wang et al.9 However, there are some aspects of these models that could represent certain limitations for the purpose of a basic understanding of the biophysics behind the cryopreservation phenomenon. A general characteristic of them is their semiempirical character. In these works the activity of the NaCl was obtained by using tabulated values.10,11 Besides that, the existence of free parameters in these nonideal models is a widespread ingredient of them. Also, the nonideality introduced by proteins is often studied together with the electrolytes; this makes it difficult to understand the role of each of these two components. Finally, these studies are usually limited to certain cooling rates or cell types;6,11 the negative aspect of this lack of generality is of course compensated by its more accurate predictive power for the studied concrete situations. In summary, they do not try to be absolutely basic approaches but satisfy the predictive goal for which they were built. The need of the present study comes from two sides. On the one hand, the need to check the extent of validity of the ideal dilute solution assumption over a range of cooling profiles, of cooling rates, and of cell types. We shall find out how sensitive this hypothesis is to the former list of variables. On the other

10.1021/jp807274z CCC: $40.75  2009 American Chemical Society Published on Web 03/13/2009

4854 J. Phys. Chem. B, Vol. 113, No. 14, 2009

Saenz et al.

hand, we have tried to fill in some gaps of the previous nonideal studies. In particular we have tried to keep loyal to the derivation of the nonideal behavior from first physical principles, as represented by the extended Debye-Hu¨ckel theory of electrolytes solutions.12 For dealing with nonideal solutions, the chemical potential of the species present in the solution must be expressed as a function of the activities or the activity coefficients, which are a function of the temperature. In the case of cryopreservation protocols the corresponding equations must be coupled with the dehydration equation. This gives rise to some more complex algebra. Nevertheless, modern computers can easily compute their solutions. As an example of application we chose mouse sperm. This is a cell type that resists accurate modeling13,14 and has become an intrinsic interest in cryopreservation and reproductive medicine. To compare this particular cell type with other interesting mammalian cells, we also study the case of mouse hepatocyte.15 This paper is organized in the following way. In section 2 we start by studying the ideal dilute solution model for a cryopreservation protocol for the sake of completeness, for nomenclature, and for the ability to make a later comparison with the nonideal case. Due to their different capability in managing the degree of supercooling, three different cooling profilesslinear, quadratic, and exponentialswill be analyzed in section 3, the ideal equation being solved and representing graphically its solution for the case of mouse sperm. In section 4 the nonideal situation is studied. In this case the vapor pressures in the Raoult law are written as a function of the solvent activity instead of the molar fractions. This activity is then related to the solute (NaCl) concentration and its activity coefficient by means of the Gibbs-Duhem equation. The Debye-Hu¨ckel theory allows us to compute the activity coefficient of this strong electrolyte as a function of its molality and the temperature of the system. In section 5 the parameters of the Debye-Hu¨ckel theory are computed. In section 6 the nonideal equation is solved and its solutions compared with the ideal case. In this section we also explain in which cases the effect of nonideality is minor or relevant, depending on the cooling profile and cell type (mouse sperm or hepatocyte). In section 7 the ideal dilute solution model is recovered from the nonideal model as a limiting case to serve as a test of the consistency of our approach.

In eq 1 the osmotic pressures can be written in terms of the vapor pressures,18

( ) ( )

* PA,l RT π ) ln in jA V P

(2)

* PA,l RT ln ex jA V P

(3)

in

πex )

* j A is the partial molar volume of the solvent, PA,l where V is in the vapor pressure of the solvent, P is the vapor pressure of the intracellular solution, and Pex is the vapor pressure of the extracellular solution. Under the dilute solution hypothesis the partial molar volume of the solvent in the solution is approximately equal to the molar volume of the pure solvent VA*. Taking this into account and introducing eq 2 and eq 3 into eq 1, we get

( )

Lp(T)ART dV Pex ln ) dt VA* Pin

(4)

We compute now the ratio between the vapor pressures in eq 4. For the intracellular solution, the ideal Raoult law allows writing Pin as * Pin ) PA,l (T)xAin + PB*(T)xBin

(5)

* (T) and P*B(T) are the vapor pressure of the pure where PA,l water and salt and xAin and xin B are their molar fractions, respectively. By neglecting the solute vapor pressure against that of the solvent and taking into account that xAin . xBin (because we are assuming dilute solution), we can write

* Pin ≈ PA,l (T)xin A

(6)

For computing Pex it is important to realize that this solution is in equilibrium with ice. Hence, it is possible to prove that Pex is approximately equal to the vapor pressure of ice (see Appendix):

2. Ideal Dilute Solution Model The analysis in this paragraph was basically done by Mazur.3 However, we develop it here for the sake of completeness, notation, and clarification of the hypothesis and for a later comparison with the nonideal case. Usually, during the cooling process in a cryopreservation protocol, extracellular ice is induced (seeded) at a given subzero temperature. This produces an imbalance between the internal and external osmotic pressures (πin and πex, respectively) that gives rise to a water flux across the cell membrane given by16,17

* Pex ≈ PA,s (T)

(7)

* Pex PA,s(T) 1 ≈ * Pin PA,l (T) xAin

(8)

From eq 6 and eq 7,

and therefore eq 4 becomes

dV ) Lp(T)A(πin - πex) dt

(1)

where V is the osmotic active volume of the intracellular solution, t is time, Lp(T) is the hydraulic conductivity of the membrane as a function of the temperature, T, and A is the cell surface area.

(( ) )

* Lp(T)ART PA,s (T) dV ln - ln xAin ) * * dt VA PA,l(T)

(9)

In this equation the dependence of the pure solid and liquid water vapor pressures on the temperature, P*A,s(T) and P*A,l(T), are given by the Clausius-Clapeyron equation18

Single-Cell Cryopreservation Protocols

J. Phys. Chem. B, Vol. 113, No. 14, 2009 4855

LV(T) d * )) (ln PA,l dT RT2

(10)

LS(T) d * )) (ln PA,s dT RT2

(( )

T

LF(T)

0

RT2

∫T

(∫

T

273.15

)

(12)

LF(T) RT2

)

dT - ln xA

(13)

In the dehydration equation, eq 13, the only independent variable we are interested in is the temperature and the only dependent variable is the volume of the intracellular solution that is osmotically active; therefore, we must express the rest of the variables as a function of them. For the change of volume with the temperature we can write:

dV dV dT dV ) ) F(T) dt dT dt dT

(14)

where F(T) is a function that depends on the cooling protocol. The molar fraction of the solvent is

xA )

nA nA + νBnB

(15)

where nA and nB are the number of moles of the solvent (H2O) and solute (NaCl) and νB the dissociation coefficient of this salt. Now we can take into account that the volume of the intracellular solution is

j A + nBV jB V ) nAV

V - nBV B∞ V - nBV B∞ + νBnBVA*

and the final dehydration equation reads

RT2 ln

(

dT -

))

V - nBV B∞ V - nBV B∞ + νBnBVA*

(18)

For the dependence of LF(T) on T we shall use the expression 3,19,20

LF(T) ) a + b(T - R) - c(T - R)2 + d(T - R)3

(19) with a ) 6004.8 J/mol, b ) 38.016 J/(mol · K), c ) 0.11088 J/(mol · K2), d ) 0.00090432 J/(mol · K3), and R ) 273.15 K. We shall study this ideal solution model in four cases: (i) thermodynamic equilibrium and (ii) linear, (iii) quadratic, and (iv) exponential cooling profiles. (i) Thermodynamic Equilibrium of the Intracellular Solution. Thermodynamic equilibrium implies three different equilibriums: mechanical, thermic, and diffusional. To achieve mechanical equilibrium, the intracellular hydrostatic pressure must equal the stress state of the membrane plus the extracellular hydrostatic pressure. The assumption is made that this equilibrium is always satisfied. In the same way, we shall assume thermal equilibrium, that is, that the temperature inside the cell is uniform and equal to that of the extracellular solution. Finally, the diffusional equilibrium requires no mass transport inside the cell and no fluxes between the cell and the exterior. Internal diffusional equilibrium is guaranteed due to the small dimensions of the cell. For the existence of diffusional equilibrium between the cell and the external media, the change in the composition inside the cell, caused by the osmosis and that produce its dehydration, must be fast enough to be able to follow the equilibrium composition that is determined by the temperature of the system and is equal to the liquid composition of the external media. The flux of water must be instantaneous, in such a way that, except in an infinitesimal lap, there is no water evacuation. This instantaneous diffusive equilibrium is guaranteed by imposing the same value of the osmotic pressure for the intra- and extracellular solutions. By applying the relationship between osmotic and vapor pressures, eqs 2 and 3, this condition reads

Pex(T) ) Pin(T)

(17)

(20)

Now, by using eqs 8, 10, 11, and 18, we find the equilibrium equation

(16)

j B is the molar partial volume of the salt, and applying where V the hypotheses of dilute solution hypothesis, that is, apj B by the molar volume of the j A by VA* and V proximating V salt at infinite dilution, V∞B , eq 15 transforms into

xA )

LF(T)

T

273.15

dT - ln xA

where we have omitted the superscript “in” in xA to simplify the notation, without the risk of confusion. At T0 ) 273.15 K the vapor pressure of the ice and liquid water are the same, so the first term in the parentheses cancels out and eq 12 transforms into

Lp(T )ART dV ) dt VA*

(∫

(11)

where LV(T) and LS(T) are the vaporization and sublimation latent heats, respectively. By integrating eqs 10 and 11 and taking into account that LS(T) - LV(T) is the fusion latent heat, LF(T), eq 9 reads * Lp(T)ART PA,s (T0) dV ln + ) * * dt VA PA,l(T0)

Lp(T)ART dV ) dT VA*F(T)

V - nBV B∞ V-

nBV B∞

+

νBnBVA*

(∫

) exp

T

273.15

LF(T) RT2

)

dT

(21)

(ii) Linear Cooling Profile. The linear cooling profile is the most widely used in cryobiology. However, it is not the optimal cooling profile, as we shall see, from the perspective of the likelihood of intracellular ice formation. Besides this, a sample never follows exactly the linear cooling rate imposed by the controlled freezer due to its finite volume. The linear cooling rate is give by


Saenz et al.

T(t) ) -Bt + T0

(22)

where T0 is the initial temperature of the system and B is a positive constant that parametrizes the whole family of linear cooling rates. The function F(T) in eq 18 is given by

F(T) )

dT ) -B dt

(23)

So, by substituting eq 23 into eq 18, the expression for the dehydration equation in this case of lineal cooling reads

Lp(T)ART dV ) * dT VA(-B)

(∫

LF(T)

T

273.15

RT2

dT -

(

ln

V - nBVB∞ V - nBVB∞ + νBnBVA*

)

However, eq 30 is always satisfied because k2 > 0 and T0 - T > 0. Therefore, by substituting eq 28 (its additive form) into eq 26, we get

F(T) )

dV ) dT

Lp(T)ART VA*(- √k12 + 2k2(T0 - T))

(

ln

(24)

(25)

dT ) -k1 - k2t dt

(26)

d2T ) -k2 dt2

(27)

From eq 26 and eq 27 it is clear that k1 is the initial cooling rate and k2 is the absolute value of the cooling acceleration (derivative of the cooling rate with respect to time). Although F(T) is the change of the temperature with time, it is in fact a function of the temperature, and not of the time. So, from eq 25 we can get the time as a function of the temperature

(28)

t must be real and positive. The radicand in eq 28 is positive because it is a sum of positive quantities: k12 > 0, k2 > 0, and T0 - T > 0, so the real character of t is guaranteed. About the positive value of t, it is necessary to take only the “+” sign in the numerator of eq 28 and to satisfy the relation

-k1 + √k12 + 2k2(T0 - T) > 0

(29)

that transforms into

k12 < k12 + 2k2(T0 - T)

(

T ∫273.15

LF(T) RT 2

dT -

V - nBVB∞

V - nBVB∞ + νBnBVA*

))

(32)

(iv) Exponential Cooling Profile. It is given by

where T0 is the initial temperature of the system and k1 and k2 are two positive constants. The physical interpretation of k1 and k2 is obtained by computing the first and second derivative of eq 25 with respect to time:

-k1 ( √k12 + 2k2(T0 - T) t) k2

(31)

So, the expression for the dehydration equation in the quadratic case is obtained by carrying eq 31 into eq 18:

(iii) Quadratic Cooling Profile. The quadratic cooling profile is given by

1 T(t) ) T0 - k1t - k2t2 2

dT ) - √k12 + 2k2(T0 - T) dt

(30)

( )

T(t) ) T0 exp -k

t T0

(33)

where T0 and k are constants. T0 is, as before, the initial temperature of the system. By taking the derivative of eq 33 with respect to the time in the origin,

dT dt

|

) -k

t)0

(34)

we see that k is the absolute value of the initial cooling rate. From the second derivative of eq 33 it is easy to realize that the cooling acceleration decreases with the temperature. As before, in order to get F(T) we have to derive eq 33 with respect to the time and express t as a function of T by using eq 33 itself. After substituting the so obtained F(T) in eq 18, we get the dehydration equation in the case of an exponential cooling profile:

)(

Lp(T)ART dV ) dT T VA* -k T0

(

T ∫273.15

LF(T) RT2

(

ln

dT V - nBVB∞ V - nBVB∞ + νBnBVA*

))

(35)

3. Dehydration and Supercooling in the Ideal Case for Different Cooling Protocols Before solving the dehydration equation for the four different cooling profiles, it is necessary to find out expressions that relate the different parameters that characterize them, to be able to compare the results. To do that, we have imposed the following conditions to the nonlinear cooling profiles: (a) The initial cooling rate for the quadratic and exponential profiles must coincide with that of the linear cooling profile which we are comparing them with. Therefore

B ) k1 ) k

(36)



(b) The initial cooling acceleration for the quadratic and exponential profiles must be the same. Therefore

k2 )

B2 T0

(37)

Equations 36 and 37 allow us to parametrize the nonlinear cooling profiles as a function of the linear cooling rate. Therefore the expressions for F(T) are the following:

F(T) ) -B;

F(T) ) -B

for a linear cooling profile

3-2

T F(T) ) -B ; T0

T ; T0

(38)

for a quadratic cooling profile

(39) for an exponential cooling profile

(40)

By substituting eqs 39 and 40 into eqs 32 and 35, respectively, we get the dehydration equations for the quadratic and exponential profiles that can be compared with the linear one (eq 24). These equations can be applied to a wide variety of cell types. We shall apply them to the interesting case of mouse sperm. For it, Lp ) 0.01 µm/(min atm) and ELp ) 94.1 kJ/mol.13 If we model the sperm by a cylinder of 0.46 µm of radius and 122 µm of height,13 the total surface area is A ) 3.5394 × 10-10 m2 and the initial osmotically active volume (61% of the total volume) is V0 ) 3.1629 × 10-17 m3. The initial value of nB is 9.014 × 10-15 mol. The resolution of the differential equations was carried out by a Pentium IV PC with the commercially available MATLAB 5.3 software.21 In particular we use the computing tool “Ode15”. Ode15 uses a method of variable order based on numerical differentiation formulas (NDF’s). In Figures 1 and 2 we display the results in a graphical way for the different profiles and cooling rates. The dashed line (--) represents the equilibrium curve; the solid lines (ss) are for the linear cooling profiles; the “plus” curves (++) are for the quadratic profiles; finally, the “diamond” curves (]]) are for the exponential profiles. Figure 1 shows the fraction of intracellular active water as a function of the temperature of the system. We see how, for a given temperature and an initial cooling rate, the quadratic profiles always keep the maximum volume of intracellular water and the exponential profile performs the maximum dehydration. The linear case always represents an intermediate situation between the quadratic and the exponential. These differences are more relevant for intermediate initial cooling rates (around 50 °C/min), being really small for high and low cooling rates. The degree of supercooling as a function of the temperature for the three studied cooling profiles is represented in Figure 2. The degree of supercooling is directly related to the likelihood of intracellular ice formation. As before, relevant differences are only present for intermediate cooling rates. The degree of supercooling is not a monotonically increasing or decreasing function for a given cooling rate. For slow and intermediate cooling rates it looks like an inverted U-shape, an indication that two competing effects are taking place: on the one hand, the driving force (difference of osmotic pressures) is stronger for lower temperatures; on the other hand, the hydraulic conductivity will decrease when the temperatures go down.

Figure 1. Representation of the intracellular volume (indirectly, the cellular dehydration) for different cooling profiles (linear, quadratic, and exponential) and several initial cooling rates in the case of the ideal dilute solution model for mouse sperm. The vertical axis represents the remaining fraction of initial active water of the cell, and the horizontal one the temperature of the system. The dashed line (--) is for the equilibrium curve, the solid line (ss) is for the linear profile, the “plus” line (++) is for the quadratic profile, and the “diamond” line (]]) is for the exponential profile. For high cooling rates (g1000 °C/min) and low cooling rates (e15 °C/min) the different profiles generate practically the same dehydration at a given temperature. However, for intermediate cooling rates the difference is notable.

Figure 2. Representation of the intracellular degree of supercooling (°C) for different cooling profiles (linear, quadratic, and exponential) and several cooling rates in the case of the ideal dilute model for mouse sperm. The dashed line (--) is for the equilibrium curve, the solid line (ss) is for the linear profile, the “plus” line (++) is for the quadratic profile, and the “diamond” line (]]) is for the exponential profile. For a given initial cooling rate and a given temperature, the linear cooling profile is always intermediate between the quadratic and the exponential.

Therefore, the effect of the temperature on the water transport does not always go in the same direction as the system cools down. After this excursion into the dilute ideal solution model, complemented with an analysis for different cooling profiles (linear, quadratic, and exponential), we shall tackle the issue of the nonideality introduced by the strong electrolyte character of sodium chloride.


Saenz et al.

4. Nonideal Dilute Solution Model Although the approximation of dilute solution could be plausible to some extent in certain cases, that of the ideal solution is not. One important reason is the strong electrolyte character of sodium chloride, which gives rise to intense ionic forces. These solute-solute interactions are present even for very high dilutions. Before proceeding, it is important to remember the differences between dilute, ideal, and ideal dilute solution.18 A solution is dilute if the sum of the amount fractions of all the solutes is small compared to 1. A solution is ideal if solvent-solvent and solvent-solute interactions are identical, so that properties such as volume and enthalpy are exactly additive. As a consequence, ideal solutions satisfy Raoult’s law. Finally, an ideal dilute solution is a dilute solution in which the solute may be regarded as obeying Henry’s law, so that all the solute activity coefficients may be approximated to 1. In other words, an ideal dilute solution is a solution that behaves ideally only for high dilutions. In the models for nonideal solutions it is necessary to distinguish between those for electrolyte and those for nonelectrolyte species. Depending on the specific application, the nonideal behavior of the solution due to the presence of nonelectrolytes can be modeled by different specific theories.18 In the present work we are concerned with the nonideality introduced by the NaCl. In this case several options are available: Debye-Hu¨ckel,12,22 Guggeheim,23 Davies,24 Scatchard,25 Pitzer,26 and Kielland27 theories among others, being only some of the versions of the Debye-Hu¨ckel, the only ones obtained from first principles and the basis for other derivations. The Debye-Hu¨ckel theory has been tested and used in a broad spectrum of branches of science and technology, such as colloids,28 quantum field theory,29 nanotubes,30 tissue engineering,31 astrophysics,32 nuclear physics,33 microfluidics,34 geology,35 and biology,36 to quote a few of them. The starting point is again eq 4, but instead of using the ideal Raoult’s law (eq 5), we should use its nonideal version: * Pin ) PA,l (T)aAin + PB*(T)aBin

(41)

where aAin and ain B are the water and salt activities, respectively. Neglecting the vapor pressure of the salt against to that of the solvent, eq 41 transforms into * * Pin ≈ PA,l (T)aAin ) PA,l (T)γAinxAin

(42)

where γAin is the activity coefficient of the intracellular solvent. For Pex eq 6 is still valid because in its deduction the ideality of the solution was not necessary (see Appendix). By inserting eq 42 into eq 4, we get

(( ) )

* Lp(T)ART PA,s (T) dV ln - ln aAin ) * * dt VA PA,l(T)

(43)

instead of eq 9. In parallel with the ideal case, this equation can be written as


(∫

T

273.15

LF(T) RT2

)

dT - ln aA

(44)

where, like before, we have omitted the superscript in aAin to simplify the notation. The activity of the solvent, aA, can be written in terms of the activity and concentration of the solute by using the Gibbs-Duhen equation.18 Due to the small volume of the cells, the characteristic time for heat transfer is several orders of magnitude lower than the characteristic time for mass transfer. Therefore, in the Gibbs-Duhem equation we can assume that T is a parameter of the system. The pressure is constant and equal to the atmospheric pressure. When applied at constant pressure and temperature to the intracellular solution, the Gibbs-Duhem equation reads

nA dµA + nB dµB ) 0

(45)

where µA and µB are the chemical potentials of the water and salt present in the cell, respectively. By writing eq 45 in terms of the molality of the solute, mB,37 and the molecular mass of the solvent, MA, it transforms into

dµA ) -mB

MA dµ 1000 B

(46)

In this equation the chemical potential of the solvent is

µA ) µA* + RT ln aA

(47)

and that of the solute:

µB ) µB* + νBRT ln ν( + νBRT ln γ( + νBRT ln mB (48) because of its strong electrolyte character. Here, ν( is the mean dissociation coefficient and γ( is the mean ionic activity coefficient. By differentiating eqs 47 and 48 under constant pressure and temperature conditions and substituting into eq 46

d ln aA ) -

MAνBmB (d ln γ( + d ln mB) 1000

(49)

Integration of eq 49 gives the activity of the water:

ln aA ) -

MAνB (m + mB ln γ( 1000 B

∫ ln γ( dmB) + C (50)

where C is an integration constant that will be calculated in the next section. To perform the integration on the right-hand side of eq 50, the activity coefficient of the salt, γ(, must be written as a function of its molality, mB. This will be done by applying the Debye-Hu¨ckel theory. There are several versions of this theory: limiting law, extended, Davies, and Pitzer theories. Historically, the extension of the original theory to other ones was done with the intention of including a broader range of electrolyte concentrations. However, only the limiting and the extended Debye-Hu¨ckel theories can be derived exclusively from first principles, the rest containing ad hoc assumptions or free parameters. The limiting law is the simplest to use. In this theory γ( is given by



ln γ( ) -z+ |z- |A(T)√Im

(51)

where z+ and z- are the electric charges of the positive and negative ions (Na+ and Cl-), A(T) is described in the next section and Im is the ionic force, defined as

Im )

∑

1 mz2 2 i)+,- i i

(52)

with mi being the molality of the solved species i. In our case mNa+ ) mCl- ) mNaCl ) mB because the salt (NaCl) is completely dissociated so no ionic pairs are present, z+ ) z- ) 1. Therefore,

Im ) mB

A(T)√Im

(54)

1 + aB(T)√Im

where a and the functions A(T) and B(T) are described and computed in the next section. The extended Debye-Hu¨ckel theory has a variable range of validity, being broader for monocharged ions, as in the present case, especially in the event of highly dielectric solvents such as water.38 For very low temperatures our approach probably will reduce its validity. However, as we shall see, the bigger deviation from the ideal dilute case happens at relatively high temperatures, when the concentration of the electrolytes is not high yet; other approaches like Pitzer’s or Davies’ theories, introduce different terms that improve the fitting to higher concentrations, but they have no theoretical justification. These models are purely empirical. Because of their simplicity, they are employed in many of the chemical equilibrium computer programs. However, their use would take us away from our intention of a first-principle derivation and would be difficult to justify if we want to keep the dilute solution hypothesis. Therefore, from eqs 53 and 54

ln γ( )

[

-A(T)√mB

(55)

1 + aB(T)√mB

After substitution of eq 55 into eq 50 we get

LF(T)

T

273.15

mB )

Carrying eq 56 into eq 44 gives us the dehydration equation in the nonideal case:

] }

nB

(58)

FA(V(T) - nBVB∞)

5. Calculation of a,A(T), B(T), and C In the extended Debye-Hu¨ckel model a is the mean ionic diameter. The radii of Na+ and Cl- are 0.95 and 1.81 Å, respectively,39 so a ) 2.76 × 10- 10 m. A(T) and B(T) are defined as

A(T) ) RT -3/2

(59)

B(T) ) βT -1/2

(60)

with

R ) (2πNAFA)1/2

β)e

(

e2 4πε0εr,Ak

( ) 2NAFA ε0εr,Ak

)

3/2

(61)

1/2

(62)

where NA is Avogrado’s number, e is the electrical charge of the electron, k is the Boltzmann constant, FA is the density of the water, ε0 is the electrical permeability of vacuum, and εr,A is the relative permeability of the water.39 Finally, the value of the integration constant C can be determined from the equilibrium curve. The molality of the solute mB and the activity of the solvent aA, at reference temperature Tref, are given by

2 2A(T) (aB(T)√mB) + ln(1 + aB(T)√mB) 2 a3B3(T)

]

( )

where nB is the number of moles of salt, FA is the water density, and V(T) is the intracellular solution volume.

[

)

dT +

It only remains to express mB as a function of the intracellular solution volume:

mB,ref )

(1 + aB(T)√mB) +mB + C (56)

RT2

ln(1 + aB(T)√mB) - (1 + aB(T)√mB) + mB - C (57)

A(T)mB√mB MAνB ln aA ) + 1000 1 + aB(T) m √ B

(

{∫

2 A(T)mB√mB MAνB 2A(T) (aB(T)√mB) + 3 3 + 1000 1 + aB(T) m 2 √ B a B (T)

(53)

However, the limiting law (eq 51) is valid only for highly diluted solutions, which is not the case of our application. So, a more appropriate expression for γ( is given by the extended Debye-Hu¨ckel theory:

ln γ( ) -z+ |z- |


nB FA(V(Tref) - nBVB∞)

aA,ref ) xA,refγA,ref ) xA,refγA(Tref)

(63)

(64)

where

xA,ref )

V(Tref) - nBVB∞ V(Tref) - nBVB∞ + νBnBVA*

(65)


Saenz et al.

Therefore, the value of the integration constant C is given by

C ) ln aA,ref +

(

[

A(Tref)mB,ref√mB,ref MAνB + 1000 1 + aB(T ) m ref √ B,ref

2A(Tref) (aB(Tref)√mB,ref)2 + ln(1 + aB(Tref)√mB,ref) 2 a3B3(T ) ref

]

)

(1 + aB(Tref)√mB,ref) + mB,ref (66) The activity coefficient at the reference temperature Tref is given by the freezing point depression curve of water, T ∫273.15

ln(γAxA) )

LF(T) dT RT

(67)

particularized at Tref:

γA(Tref) )

∫T xA,ref 273.15 1

ref

LF(T) dT RT

(68)

where LF(T) is given by eq 19. By integrating eq 68, we get the activity coefficient of the water at temperature Tref:

[

(

)

1 a - bR - cR2 - dR3 1 1 + xA(Tref) R 273.15 Tref Tref b + 2Rc + 3R2d c + 3Rd ln (Tref - 273.15) + R 273.15 R d (T 2 - 273.152) (69) 2R ref

γA(Tref) )

(

)

]

6. Dehydration and Supercooling in the Nonideal Case in Comparison with the Ideal Case Similarly to the ideal case, the nonideal one for linear, quadratic, and exponential profiles was studied. We shall make use again of the relations in eqs 36 and 37 in order to be able to compare different types of profiles. Therefore, the function F(T) is given by the eqs 38-40. In Figure 3 the cell dehydration in the case of the nonideal solution model (circles) for different linear cooling rates is given. At first glance the results are very close to the ideal case, also plotted in the same graphics by the solid lines. This is also the case for the quadratic and exponential profiles. So, the first conclusion is that the approximation of ideal dilute solution is adequate in many applications. However, from a more detailed analysis such as the one shown in Tables 1-3, it is possible to see the existence of important differences at relevant temperatures. In these tables we show the highest deviation of the ideal with respect to the nonideal case for different linear cooling rates, as well as the temperature for which this highest deviation happens, for mouse sperm. The relative error (rel error) has been computed as

rel error )

|

|

ideal vol - nonideal vol × 100 nonideal vol

(70)

where ideal vol means the active cell water volume predicted by the ideal model and nonideal vol means the active cell water volume predicted by our nonideal model.

Figure 3. Comparison between the ideal dilute and nonideal (Debye-Hu¨ckel based) solution models. The dashed line (--) is for the equilibrium curve, the solid line (ss) is for the linear profile, the “plus” line (++) is for the quadratic profile, and the “diamond” line (]]) is for the exponential profile. The upper equilibrium curve is for the ideal dilute solution model and the lower one for the nonideal. The correction introduced by the use of the nonideal solution model represents, in all the cases, a higher dehydration for a given initial cooling rate and a given temperature. This implies a higher concentration of salts inside the cell and a lower degree of supercooling (and, therefore, a lower likelihood of intracellular ice formation with respect to the one based in the ideal dilute solution assumption). In the inset two relevant cooling rates (30 and 40 °C/min) are magnified. Differences in the degree of dehydration between ideal and nonideal models of almost 20% are predicted at these relevant cooling rates.

TABLE 1: Differences in the Degree of Dehydration (% (v/v); See Text) Predicted by the Ideal Dilute Solution Model and the Nonideal One for Selected Relevant Linear Cooling Rates in the Case of Mouse Sperma cooling rate max rel (°C/min) error (%) -5 -10 -15 -20 -25 -30

13.8428 14.6054 15.0762 15.2829 15.5559 16.5138

temp (°C) -6.0996 -8.2910 -10.9022 -12.6966 -16.2375 -18.5490

cooling rate max rel (°C/min) error (%) -35 -40 -45 -50 -55 -60

17.9140 8.1526 2.5174 0.9224 0.4073 0.3883

temp (°C) -62.3133 -66.8282 -47.2994 -43.0053 -10.1218 -10.3766

a The maximum error (17.9%) happens at a cooling rate of -35 °C/min and at relatively low temperatures (-62.3 °C).

In Table 1 we can see that the relative error is small for the highest cooling rates. In this case the dehydration is low, so the extended Debye-Hu¨ckel model can keep its accuracy in a big range of temperatures. In the case of low cooling rates we see that there is a significant (more than 13%) deviation even at relatively high temperatures, when there is not a high dehydration, guaranteeing the validity of the model. These conclusions can also be drawn from Figure 4, where the dependence of this error is represented with respect to the cooling rate at a given temperature. Only in the range of cooling rates represented in this figure is the difference notable. This range is inside the interval of cooling rates commonly applied in many cryopreservation protocols for cell suspensions. Interestingly, there is a high reduction in this error for cooling rates beyond -40 °C/min. This fact is better observed in Figure 5. The case of exponential cooling profiles is shown in Table 2. The maximum relative error also happens at -35 °C/min, but in this case the temperature of the system is only -28 °C.



TABLE 2: Representation of the Differences in the Degree of Dehydration (% (v/v); See Text) Predicted by the Ideal Dilute Solution Model and the Nonideal One in the Case of Exponential Cooling Profiles for Mouse Sperma initial cooling max rel rate (°C/min) error (%) -5 -10 -15 -20 -25 -30

13.7403 14.6196 15.0465 15.1915 15.7050 15.8304

temp (°C) -6.2542 -8.1832 -10.2680 -12.7900 -15.5842 -19.6434

initial cooling max rel rate (°C/min) error (%) -35 -40 -45 -50 -55 -60

16.5189 12.5283 4.0886 1.4984 0.6754 0.3902

temp (°C) -28.0313 -83.0000 -62.6293 -40.3483 -41.1572 -10.4491

a In this case the situation is somewhat different with respect to the linear profiles: for the same initial cooling rate; the relevant differences happen at higher temperatures.

TABLE 3: Representation of the Differences in the Degree of Dehydration (% (v/v); See Text) Predicted by the Ideal Dilute Solution Model and the Nonideal One in the Case of Mouse Hepatocytea initial cooling max rel rate (°C/min) error -0.5 -1 -1.5 -2 -5 -10

10.8916 11.1585 11.0410 10.7572 8.1113 1.7157

temp (°C) -8.4839 -10.1157 -12.1499 -14.1385 -33.3640 -79.0069

initial cooling max rel rate (°C/min) error -15 -20 -25 -30 -35 -40

0.9789 0.6738 0.5039 0.4338 0.3609 0.2927

temp (°C) -76.4792 -81.0748 -54.6808 -80.4774 -77.2275 -55.7208

a

It is introduced to show the different behavior with respect to the different mammalian cells, particularly when compared with the formerly studied case of mouse sperm. This comparison makes sense due to the particular geometry of the gametes. We see how in the case of mouse hepatocytes, the cooling rates for which the significant differences between the ideal dilute model and the nonideal one happen are much lower. For cooling rates in the range of -1 to -5 °C/min (typical cooling rates for these cells), the error is around 10% and happens around -10 °C.

Figure 5. Dependence of the highest deviation of the ideal dilute model with respect to the nonideal one with the cooling rate in the case of linear profiles for mouse sperm. The error increases very steady up to -30 °C/min. Then, a drastic reduction of this deviation happens, being insignificant for cooling rates higher than -55 °C/min.

To see how mouse sperm compares with other typical mammalian cells, and to stress the difference of sperm when it undergoes cryopreservation protocols, we show in Table 3 the maximum errors between ideal and nonideal solution models in the case of mouse hepatocytes. We see that the cooling rates for which an important deviation occurs is much lower. For typical cooling rates of these cells (1-5 °C/min) in standard cryopreservation protocols15 we see that the error is still 11.16% and happens at relatively high temperatures (-10.16 °C). 7. Recovery of the Ideal Case as a Limit of the Nonideal One As a consistency test, we shall recover the ideal dilute solution model as a limiting case of the nonideal one in the following two steps: (i) identify the condition of ideality; (ii) substitute such a condition into the nonideal model and compare the obtained result with the ideal case. (i) A dilute solution is considered ideal if the activity of the solvent is 1 (γA ) 1 in eq 44). In our case this is confirmed by a simple inspection of eqs 13, 42, and 44. In section 3 we wrote the activity of the solvent aA as a function of the activity and concentration of the solutes (eq 49). By performing similar steps for the case of the ideal solution, we get

Figure 4. Deviation of the ideal dilute model with respect to the nonideal one for different linear cooling rates for mouse sperm. We see that for intermediate cooling rates the deviation is more important. From these graphs it is evident that the behavior of this error with the temperature is different for low than for intermediate cooling rates. In the first case, an inverted U-shape behavior is displayed, while in the case of intermediate cooling rates the error keeps quite constant with the temperature (once the maximum is reached).

An important second difference between the linear and the exponential cooling profiles is that in the exponential case the smallest errors (below 1%) are found at highest cooling rates.

d ln xA ) -

MAνBmB d(ln mB) 1000

(71)

By comparing eqs 49 and 71, we see that whenever γ( ) 1 the activity of the intracellular solvent and its molar fraction coincide, so γA ) 1. Hence, we can impose γ( ) 1 on the Debye-Hu¨ckel theory as being the limiting case to recover the ideal dilute solution model. Taking the limit γ( ) 1 in the expression of the mean ionic activity coefficient, eq 55, we find that A(T) ) 0 for any value of T.


Saenz et al.

(ii) Now, by assuming A(T) ) 0 in the eq 56 and writing the molality of the solute as a function of the intracellular volume (eq 58), we find


{

T ∫273.15

LF(T) RT2

dT +

νBnBVA* V(T) - nBVB∞

}

- C0

(72)

where

C0 ) ln aA,ref +

νBnBVA* V(Tref) - nBVB∞

(73)

Under the hypothesis of dilute solution (assumed in all this work)

νBnBVA* V - nBVB∞

,1

(74)

and the molar fraction of the solvent, is given by

(

ln xA ) ln

V - nBVB∞ V - nBVB∞ + νBnBVA*

(

) - ln 1 + ≈



)

) (75)

After substitution of the last term of eq 75 in eq 73 we find that C0 ) 0 in eq 72. Once eq 75 is carried also to eq 18 (dilute solution hypothesis), we realize that eq 72 and eq 18 are in fact identical. So, under the hypothesis of dilute solution, the ideal model is recovered as a limiting case of the nonideal one, confirming the consistency of this analysis. 8. Discussion Models for cell dehydration are the basis for the design of slow freezing cryopreservation protocols. A good model should be able to predict the optimal cooling profile and cooling rate in order to avoid intracellular ice formation. A proper choice of the optimal cooling profile and cooling rate can reduce dramatically the amount of cryoprotectant to be used, lowering the toxic effects of this chemical on them or on the tissue obtained from their culture. Even though vitrification is emerging as a very promising technology, especially in the cases where low cryoprotectant concentration is used,40 slow freezing is still the most used technique, and the one with the widest range of applicability. The theoretical formulation of these models started with Mazur3 and relies on a set of hypotheses that are usually satisfied to some extent. With this work we contribute to the exploration of one of these hypotheses, that of the ideality of the solutions. An ideal solution is an approximation based on the assumption that the molecules of solute only interact with those of the solvent. This is particularly valid for nonstrong electrolyte species. But if strong electrolytes are present, such

as NaCl, then there are solute-solute interactions even in the case of very dilute solutions. There are different possibilities for exploring the nonideality of the solution. The approach followed by other authors (Levin, Hunt, Arnoud, and Pegg, etc.) is based on experimental Values for the activity coefficients.6-11 This path, however, limits the possibility of a deeper understanding of the physical phenomena under these models. We have preferred to use fundamental tools, such as the Debye-Hu¨ckel extended theory,12 which opens the possibility of building sometime an entire theory from very basic physical principles. The changes produced by the ideality are minor or relevant depending on several factors. (i) If these changes take place at higher temperatures (when the metabolism is still active), then their effects are more relevant than at lower temperatures. (ii) Also, if these changes happen for cooling rates around the optimal one in a typical slow freezing protocol (for a concrete cell type), then they are also more relevant than if they happen at other (not optimal) cooling rates. (iii) Finally, there are very sensitive cells (such as oocytes or stem cells) for which even minor changes produce very different results after cryopreservation. An important conclusion of our work is that the hypothesis of ideal solution is very adequate for high cooling rates. This agrees with Levin’s conclusion;6 he explored the nonideality in the range of 5000 °C/min for erythrocytes, finding only a little difference between ideal and nonideal models. Ideal models, being much easier to formulate and solve, indicate that they are more convenient to use in such high cooling rate regimes (error below 1%). However, “slow freezing” is usually “slow”. This means that there are many cell types with an optimal cooling rate that is between 0.1 and 10 °C/min. For these cooling rates though there is an important deviation of the ideal model with respect to the nonideal one. This deviation reaches up to 18% in the studied case of mouse sperm. As was to be expected, differences between different cell types are present, and we made them explicit in our work. Two important cases were compared: mouse hepatocyte and mouse sperm. In the case of mouse sperm the bigger differences between the ideal and the nonideal model are found at moderate cooling rates (35 °C/min) and low temperatures (-62 °C). At these low temperatures however, the water motility is so small that this difference is probably not worth worrying about. However, for mouse hepatocyte, the bigger differences (more than 11%) are found at around 1 °C/min, and at a high temperature (-10 °C). Due to the importance of a proper cryopreservation of this cell type for many biotechnological applications, this finding can be relevant for the design of proper freezing protocols in this case. For the sake of completeness, a comparison of different cooling profiles (linear, quadratic, exponential, and the equilibrium curve) was also shown in the paper, the linear protocol being the most studied, but not necessarily the optimum, nor the one most present in the real experiment situation, due to the size of the sample. We hope that our approach is of use to further theoretical developments, as well as in making the experimental observations compatible with the results predicted by the theory in cases where it is not totally accomplished nowadays. Acknowledgment. We thank Dr. Heidi Elmoazzen for a critical reading of the manuscript and very helpful comments on its content.



Appendix

References and Notes

In the extracellular solution

(1) Polge, C.; Smith, A. U.; Parkes, A. S. Revival of spermatozoa after vitrification and dehydration at low temperatures. Nature 1949, 164, 666. (2) Lovelock, J. E.; Bishop, M. W. H. Prevention of freezing damage to living cells by dimethyl sulphoxide. Nature 1959, 183, 1394–1395. (3) Mazur, P. Kinetics of water loss from cells at subzero temperatures and the likelihood of intracellular freezing. J. Gen. Physiol. 1963, 47, 347– 369. (4) Karlsson, J. O. M.; Toner, M. Long term storage of tissue by cryopreservation: critical issues. Biomaterials 1996, 17, 243–256. (5) McGrath J. J. Preservation of biological material by freezing and thawing. In Heat Transfer in Medicine and Biology. Schitzer, A., Eberhart, R. C., Eds.; New York: Plenum: 1985. 185-233. (6) Levin, R. L.; Cravalho, E. G.; Huggins, C. E. Effect of solution non-ideality on erythocyte volume regulaion. Biochim. Biophys. Acta 1977, 465, 179–190. (7) Arnaud, F. G.; Pegg, D. E. Permeation of glycerol and propane1,2-diol into human platelets. Cryobiology 1990, 27, 107–118. (8) Hunt, J. C.; Armitage, S. E.; Pegg, D. E. Cryopreservation of umbilical cord blood. 1. Osmotically inactive volume, hydraulic conductivity and permeability of CD34+ cells to dimethyl sulphoxide. Cryobiology 2003, 46, 61–75. (9) Wang, X.; Wexler, A. S. The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism. Bull. Math. Biol. 1994, 56, 515–546. (10) Hamer, W. J.; Wu, Y. C. Osmotic coefficients and mean activity coefficients of univalent electrolytes in water at 25 °C. J. Phys. Chem. Ref. Data 1972, 1, 1047–1099. (11) Pegg, D. E.; Hunt, C. J.; Fong, L. P. Osmotic properties of the rabbit corneal endothehum and their relevance to cryopreservation. Cell Biophys. 1987, 10, 169–191. (12) Debye, P.; Hu¨ckel., E. Zur theorie der elektrolyte. I. Gefrierpunktserniedrigung und verwandte erscheinungen. Phys. Z. 1923, 24, 185–206. (13) Devireddy, R. V.; Swanlund, D. J.; Roberts, K. P.; Bischof, J. C. Subzero water permeability parameters of mouse spermatozoa in the presence of extracellular ice and cryoprotective agents. Biol. Reprod. 1999, 61, 764–775. (14) Devireddy, R. V.; Fahrig, B.; Godke, R. A.; Leibo, S. P. Subzero water transport characteristics of boar spermatozoa confirm observed optimal cooling rates. Mol. Reprod. DeV. 2004, 67, 446–457. (15) Sugimachi, K.; Roach, K.L.; Rhoads, D. B.; Tompkins, R.G.; Toner, M. Nonmetabolizable glucose compounds impart cryotolerance to primary rat hepatocytes. Tissue Eng. 2006, 12, 579–588. (16) Jacobs, M. H.; Stewart, D. R. A simple method for the quantitative measurement of cell permeabiIity. J. Cell. Comp. Physiol. 1932, 1, 71–82. (17) Kedem, O.; Katchalsky, A. Thermodynamic analysis of the permeability of biological membranes to noon-electrolytes. Biochim. Biophys. Acta 1958, 27, 229–246. (18) Tester J. W.; Modell. M. Thermodynamics and Its Applications, 3rd ed.; Prentice Hall: New York, 1997. (19) Dorsey, N. E. Properties of Ordinary Water Substance; American Chemical Society Monograph Series No. 81; Reinhold: New York, 1940. (20) Glasstone, S. Textbook of Physical Chemistry, 2nd ed.; D. Van Nostram: New York, 1946. (21) MATLAB; The MathWorks: Natick, MA. (22) Hu¨ckel, E. The theory of concentrated, aqueous solutions of strong electrolytes. Phys. Z. 1925, 26, 93–147. (23) Guggenheim, E. A. The specific thermodynamic properties of aqueous solutions of strong electrolytes. Philos. Mag. 1935, 19, 588. (24) Davies, C. W. Ion Association; Butterworths: London, 1962. (25) Scatchard, G. The interpretation of activity and osmotic coefficients. The Structure of Electrolyte Solutions; John Wiley & Sons: New York, 1959; Chapter 2. (26) Pitzer, K. S. Electrolyte theory - Improvements since Debye and Hu¨ckel. Acc. Chem. Res. 1977, 10, 371–377, 1977. (27) Kielland, J. Individual activity coefficients of ions in aqueous solutions. J. Am. Chem. Soc. 1937, 59, 1675–1678. (28) Larsen, A. M.; Grier, D. G. Like-charge attractions in metastable colloidal crystallites. Nature 1997, 385, 230–233. (29) Samuel, S. Grand partition function in field theory with applications to sine-Gordon field theory. Phys. ReV. D 1978, 18, 1916–1932. (30) Wang, S.; Humphreys, E. S.; Chung, S. Y.; Delduco, D. F.; Lustig, S. R.; Wang, H.; Parker, K. N.; Rizzo, N. W.; Subramoney, S.; Chiang, Y. M.; Jagota, A. Peptides with selective affinity for carbon nanotubes. Nature 2003, 2, 196–200. (31) Bathe, M.; Rutledge, G. C.; Grodzinsky, A. J.; Tidorz, B. A coarsegrained molecular model for glycosaminoglycans: Application to chondroitin, chondroitin sulfate, and hyaluronic acid. Biophys. J. 2005, 88, 3870– 3887. (32) Jung, Y.-D.; Yang, K.-S. Classical electron-ion Coulomb Bremsstrahlung in weakly coupled plasmas. Astrophys. J. 1997, 479, 912–917.

ex Pex ) PA,l + PBex

(76)

By neglecting the vapor pressure of the solutes with respect to the solvents, ex Pex ) PA,l

(77)

Because the extracellular solution is always in thermodynamic equilibrium with the ice, its chemical potential has the same value:

ex * * µA,l ) µA,s (T) ) µA,g (T) ) µA0 + RT ln

( ) * PA,s

P0

(78)

where µA0 is the chemical potential at a reference pressure, P0, * is the vapor pressure of the ice at temperature T. We and PA,s can suppose that in the interface between the solution and its vapor there is not ice. Therefore, the vapor pressure is due exclusively to the liquid solvent and the solute. Because the liquid-vapor system is in equilibrium, the chemical potential of the solvent in the solution equals the chemical potential of the solvent in the vapor: ex µA,l (T) ) µA,g(T)

(79)

The solvent in the vapor phase can be modeled by the simplifying assumption of an ideal mixture of ideal gases, and, therefore, its chemical potential can be written as

µA,g(T) ) µA0 + RT ln

( ) PA,l

(80)

P0

and, therefore, substituting eq 80 into eq 79, we get

ex µA,l (T) ) µA0 + RT ln

( ) PA,l

(81)

P0

where PA,l is the partial pressure of the vapor in equilibrium with the extracelullar solution. By identifying the expressions of the chemical potential of the solvent obtained in eq 81 and eq 78, we get

µA0 + RT ln

( ) PA,l P0

) µA0 + RT ln

( ) * PA,s

P0

(82)

and from it we arrive at the expression that we were looking for: ex * Pex ) PA,l (T) ) PA,s (T)

(83)

4864 J. Phys. Chem. B, Vol. 113, No. 14, 2009 (33) Kidder, R. E.; deWitt, H. E. Application of a modified DebyeHu¨ckel theory to fully ionized gases. Plasma Phys. 1961, 2, 218–223. (34) Conlisk, A. T. The Debye-Huckel approximation: Its use in describing electroosmotic flow in micro- and nanochannels. Electrophoresis 2005, 26, 1896–912. (35) Phillips, G. N.; Evans, K. A. Role of CO2 in the formation of gold deposits. Nature , 429, 860–863. (36) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. Colloquium: The physics of charge inversion in chemical and biological systems. ReV. Mod. Phys. 2002, 74, 329–345.

Saenz et al. (37) Denbigh, K. G. The Principles of Chemical Equilibrium, 4th ed.; Cambridge University Press: Cambridge, U.K., 1981; ISBN0521281504. (38) Garrels, R. M.; Christ, C. L. Solutions, Minerals and Equilibria; Freeman,Cooper: San Francisco, 1965. (39) Lide, D. R. Handbook Chemistry and Physics, 85th ed.; Chemical Rubber: Boca Raton, FL, 2004. (40) Risco, R.; Elmoazzen, H.; Doughty, M.; He, X.; Toner, M. Cryobiology 2007, 55, 222–229.

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Comparison between Ideal and Nonideal Solution Models for Single

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