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. 2025 Feb 11;129(7):2013–2029. doi: 10.1021/acs.jpcb.4c03915

Predictive Thermodynamics for Isochoric (Constant-Volume) Cryopreservation Systems

Julia H Grenke 1, Janet A W Elliott 1,*
PMCID: PMC11849685  PMID: 39932989

Abstract

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Cryopreservation is the preservation and storage of biomaterials using low temperatures. There are several approaches to cryopreservation, and these often include the use of cryoprotectants, which are solutes used to lower the freezing point of water. Isochoric (constant-volume) cryopreservation is a form of cryopreservation that has been gaining interest over the past 18 years. This method utilizes the anomalous nature of water in that it expands as it freezes. The expansion of ice on freezing is used to induce a pressure in the system that limits ice growth. In this work, we use Gibbsian thermodynamics, the Elliott et al. multisolute osmotic virial equation, the Feistel and Wagner correlation for ice Ih, and the Grenke and Elliott correlation for the thermodynamic properties of liquid water at low temperatures and high pressures to predict how the pressure, volume fraction of ice, and solute concentration in the unfrozen fraction change as the solution is cooled isochorically. We then verified our model by predicting experimental results for saline solutions and ternary aqueous solutions containing NaCl and organic compounds commonly used as cryoprotectants: glycerol, ethylene glycol, propylene glycol, and dimethyl sulfoxide. We found that our model accurately predicts experimental data that were collected for cryoprotectant concentrations as high as 5 M, and temperatures as low as −25 °C. Since we have shown that our liquid water correlation, on which this work was based, makes accurate predictions to −70 °C, as long as the pressure is not higher than 400 MPa, we anticipate that the prediction methods presented in this work will be accurate down to −70 °C. In this work we also modeled how sealing the isochoric chamber at room temperature versus at the nucleation temperature impacts isochoric freezing. The prediction methods developed in this work can be used in the future design of isochoric cryopreservation experiments and protocols.

1. Introduction

An emerging area in the field of cryobiology is that of isochoric (constant-volume) cryopreservation. In general, cryopreservation is the preservation of live cells, tissues, and organs using extremely cold temperatures (such as that of liquid nitrogen, −196 °C). Historically, most cryopreservation has been conducted under isobaric (constant-pressure) conditions, specifically at atmospheric pressure due to it being the natural environment on earth. In isobaric cryopreservation, various solutes are used as cryoprotectants in solutions to prevent freezing injury to cells. This method is effective for some biological samples, including most single-cell-type suspensions. However, for some cell types and most multi-cell-type tissues and organs, cryopreservation protocols with sufficient post-thaw cellular viability and function have not yet been found.1

Isochoric cryopreservation provides a promising approach because it takes advantage of the anomalous nature of water where the specific volume of solid ice Ih is higher than that of liquid water. In order for the freezing of water to occur, it must expand to a higher volume. In isochoric cryopreservation, the solution is prevented from expanding as it is cooled which leads to a pressure increase.1 As the system is cooled, the volume of the liquid decreases allowing a fraction of ice to grow. However, the amount of ice formed at a given temperature is less than in isobaric cryopreservation.1 Using isochoric cryopreservation, less cryoprotectants are required because they are used in conjunction with pressure to depress the freezing point.2

Rubinsky et al. were the first to propose isochoric cryopreservation in 2005,2 and they developed thermodynamic theory to support their proposal. They used existing correlations for pure water and ice Ih created by Ter Minassian et al.3 and Nagornov and Chizhov,4 respectively. Rubinsky et al.2 assumed that the total freezing point depression of the solution could be represented by the summation of the temperature depression caused by the pressure increase and the temperature depression caused by the solutes. Their model made accurate predictions of experiments in which physiological saline or 1 or 2 M solutions of ethylene glycol or glycerol in physiological saline were isochorically frozen to approximately −12 °C. In 2019, Powell-Palm et al.5 proposed a multiphase method of isochoric freezing which was modeled using the same thermodynamic assumptions as those of Rubinsky et al.2 The data were collected for a water–NaCl–glycerol system until approximately −18 °C, and their model was able to accurately provide predictions.5 Rubinsky et al. recognized that “at higher pressures and concentrations, the effects may not be linearly additive”, and that “nonlinear correlations will probably be obtained later” when “experimental data becomes available”;2 however, no such nonlinear correlations have yet been developed. A second model that exists is a thermomechanics model by Solanki and Rabin.6 This model incorporates thermodynamics, heat transfer, fluid mechanics and solid mechanics; however, it is currently limited to pure liquid water.6 It was able to accurately predict the behavior of pure liquid water under isochoric freezing until approximately −50 °C and 400 MPa.

The two existing models by Rubinsky et al.2 and Solanki and Rabin6 have limitations that prevent them from predicting the pressure increase of complex solutions (such as those with three or more solutes) or systems with much higher concentrations of solutes.

Multiple experiments have been conducted to collect data that validate the principles of isochoric freezing that have not yet been predicted or modeled.7−10 These experiments did not include biological material, but instead consisted of possible isochoric cryopreservation solutions.

Isochoric cryopreservation has also been investigated experimentally using biological samples. Studies done by Mikus et al. on the nematode Caenorhabditis elegans,11 Powell-Palm et al. on Escherichia coli bacteria,12 Preciado and Rubinsky on kidney epithelial cells,13 Wan et al. on rat hearts,14 and Powell-Palm et al. on pancreatic islets15 have all investigated isochoric cryopreservation using a pure saline solution. The primary conclusions from these investigations were that there is a pressure limit that cells/tissues/organs can withstand during storage, as well as a time-limit of storage before the effects of the cryopreservation are detrimental. In 2019, Wan et al. continued their investigation on rat hearts but incorporated 1 M glycerol.16 They discovered that this doubled the survival time of the rat hearts. In 2021, Powell-Palm investigated a version of isochoric cooling called supercooling.17 During supercooling, the nucleation of ice is not initiated, and the solution remains liquid even below its freezing point. Investigations on isochoric supercooling of cardiac microtissues17 and pig livers18,19 also show promise. Currently, isochoric freezing has also been making significant progress in the food preservation and storage industry. Examples include the preservation of orange juice,20 pomegranate juice,21 carrot juice,22 bovine milk,23 sweet cherries,24 apples,25 pomegranates,26 blueberries,27 sweet potatoes,25 and potatoes.28

The ability to thermodynamically model isochoric cryopreservation will be extremely valuable for the future design of new experiments and protocols. For example, for cell, tissue, or organ preservation, each cell type is unique; our model would be able to aid in the design of protocols that stay within the pressure limit and toxicity limit of a specific cell type (or multiple cell types).

The objectives of this study were to use Gibbsian thermodynamics29,30 and the Elliott et al. multisolute osmotic virial equation31−35 to predict the thermodynamic behavior of solutions cooled isochorically (under constant-volume conditions). Our model incorporates the Feistel and Wagner correlation36 for ice Ih (also given in the Supporting Information of Grenke and Elliott37) and the Grenke and Elliott correlation for liquid water37 to represent the thermodynamic properties at low temperatures and high pressures. This work builds on the work by Grenke and Elliott,37 incorporating solutes into the same framework. To verify our model, we used experimental results from literature for: (i) 0.9 wt % NaCl in water (physiological saline),2,9 (ii) 1, 2, 3, 4, and 5 M glycerol (GLY) in physiological saline,2,9 (iii) 1, 2, 3, 4, and 5 M ethylene glycol (EG) in physiological saline,2,10 (iv) 1.5 M dimethyl sulfoxide (DMSO) in distilled water,7 and (v) 1, 2, 3, and 4 M dimethyl sulfoxide (DMSO) in physiological saline.10 We made predictions for how the pressure, volume fraction of ice, and the concentration of each solute in the unfrozen fraction increase as the temperature decreases. We also made these predictions for 1, 2, 3, 4, and 5 M propylene glycol (PG) in physiological saline. A secondary objective was to investigate how sealing the isochoric chamber at room temperature (298.15 K) versus at the nucleation temperature impacts the predicted pressure, volume fraction of ice, and solute concentration in the unfrozen fraction.

2. Theory

This section describes the methods used to predict (i) the pressure increase during isochoric cooling of a multicomponent solution, as well as (ii) the volume fraction of ice in the system, and (iii) the solute concentration in the unfrozen fraction. We begin with a general description of the isochoric system, then move into general equations used that govern the system and follow with molecular species-specific information and equations for the components that we focus on in this study.

2.1. General Isochoric Freezing System

The general isochoric freezing procedure described in the literature2,7,9,10 begins with the mixing of the multicomponent solution of interest at the mixing temperature, Tinitial, shown in Figure 1a. Depending on the experimental method reported in literature, the isochoric chamber is sealed at different temperatures either at or below Tinitial; therefore, we denote the sealing temperature as Tseal, shown in Figure 1b. The isochoric chamber is sealed such that there is no air present in the chamber. Because this chamber is an isochoric chamber, its volume does not change throughout cooling. Once the system is at (or slightly below) the nucleation temperature, Tnuc, ice is either manually or spontaneously nucleated (depending on the experimental protocol), which is shown in Figure 1c. What is interesting about the isochoric system is that it takes advantage of water’s anomalous freezing property, in that water expands as it freezes. The isochoric chamber restricts this expansion, leading to a pressure increase in the chamber. This pressure is recorded using a pressure transducer. Once the isochoric chamber is sealed, the volume must remain the same. Because no material is added or removed from the system, the molar volume (and density) of the entire system must also remain the same. Because the molar volume of liquid water will decrease with a decrease in temperature (and will decrease with an increase in pressure), and the molar volume of ice is higher than that of liquid due to the anomalous nature of ice Ih, in order to maintain constant volume, some ice will form. In a multicomponent solution, the ice fraction is composed of pure water; therefore, the concentration of solutes in the unfrozen fraction increases with further cooling, which is shown in Figure 1d (the higher concentration in the unfrozen fraction being illustrated with a darker blue color).

Figure 1.

Figure 1

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General isochoric freezing system (not to scale). (a) Tinitial is the temperature at which the multicomponent solution is mixed. The solute concentration of the solution mixed at Tinitial is labeled as xi,initial. (b) Tseal is the temperature at which the isochoric chamber is sealed. (c) Tnuc is the temperature at which ice is nucleated. (d) Once ice forms, the concentration of solutes in the unfrozen fraction, xi, increases (illustrated with a darker blue color). Illustration conceptually based on Powell-Palm et al.15

2.2. General Governing Equations

The first step is to predict the equilibrium freezing-point pressure for each temperature to which the system is cooled. Because of the complexity of the equations and the numerical methods, it is more straightforward to predict the equilibrium freezing-point temperature at each pressure, which is thermodynamically equivalent. This is executed by finding the temperature that satisfies the equilibrium conditions between the multicomponent liquid solution phase and pure ice Ih at each pressure. The framework of this model is Gibbsian thermodynamics,29,30 and the method used is an extension of the work done by Grenke and Elliott.37 As in our previous work,37 we assume that ice Ih is the only ice present since the pressure rise experimentally reported in the literature results from the presence of ice Ih that expands upon freezing and there exists a good correlation for the thermodynamic properties of ice Ih.36 We assume the ice Ih to be pure; to do otherwise would require knowledge of the thermodynamics of incorporation of specific solutes into ice Ih, information we do not have. We neglect any impact of surface tension so that we assume that the liquid and solid phases are at the same pressure at equilibrium (the role of surface tension could be used to extend this work, were enough properties known, by following other Gibbsian thermodynamics work38−40). We neglect any other changes of the bulk phases caused by the presence of the interface. Finally, we assume that pure ice Ih is the only solid (the role of other precipitates could be used to extend this work, were enough properties known, by following other Gibbsian thermodynamics work39,41). Therefore, the equilibrium between the two phases is defined by setting the chemical potential of the solvent (water) in the liquid and ice phases equal to one another, with the dependencies of the chemical potentials shown in parentheses

2.2. 1

where, μL1 is the chemical potential of the pure water solvent 1 in the liquid, L; μS1 is the chemical potential of the solvent 1 in the solid ice phase, S; T is the absolute temperature; P is the absolute pressure; x is the component mole fraction; and the subscripts “1, 2,...,r” represent the component number (solvent: 1 and solutes: 2, 3,...,r). It is important to note that the solid ice phase consists of only pure water; the solutes will not freeze but will instead concentrate in the liquid fraction. Thus, μS1(T,P) = μS01(T,P), with superscript 0 indicating pure species. The chemical potential of the solvent in the liquid, μL1, is described using the following equation30

2.2. 2

where μL01 is the chemical potential of the pure solvent (not in solution) as a liquid, L; R is the universal gas constant; and Inline graphic is the osmole fraction—defined as the osmolality multiplied by the molar mass of the solvent 1, Inline graphic. The equation for μL01 was derived by Grenke and Elliott,37 and is shown below

2.2. 3

where Tref and Pref are the temperature and pressure at a reference state; v̲ is the molar volume; s̲ is the molar entropy; Inline graphic is the partial derivative of molar volume with respect to temperature holding pressure constant, Inline graphic; and Inline graphic is the molar constant pressure heat capacity. Substituting eq 3 into eq 2 gives

2.2. 4

The osmole fraction can be calculated using the mole fraction version of the Elliott et al. multisolute osmotic virial equation (eq 5)32

2.2. 5

with the most recent combining rules from Binyaminov and Elliott33 (eqs 6 and 7)

2.2. 6
2.2. 7

where k+ is a fitting parameter, referred to as the dissociation constant of electrolyte i, j, or k, and B+ and C+ are the mole-fraction-based second and third osmotic virial coefficients, respectively, for solutes i, j, or k. For this problem, the mole fractions in the liquid are functions of the mole fraction of ice, y. This is because as the solution cools (temperature decreases) and the pressure increases, the fraction of ice increases. As the ice fraction increases, the concentration of solutes in the liquid fraction increases; thus, the concentration of solutes in the unfrozen liquid fraction is a function of ice fraction. Because the solutes only concentrate in the liquid fraction, the mole fraction, xi, is defined as

2.2. 8

where, ni is the moles of solute present (i = 2, 3,...,r), and nL is the total number of moles in the liquid fraction Inline graphic. Next, the molar ice fraction is given by

2.2. 9

where nS is the number of moles in the pure water ice fraction, and ntot is the total number of moles in the system (ntot = nL + nS). Rearranging eq 9 for nL gives

2.2. 10

Substituting eq 10 into eq 8 gives

2.2. 11

The total number of moles of each solute ni does not change in the system, nor does the total number of moles in the entire system ntot; therefore

2.2. 12

where xi,initial is the concentration of the solute in the solution initially mixed. Utilizing eq 12, eq 11 becomes

2.2. 13

Substituting eq 13 into eq 5 yields

2.2. 14

Next, the equation that represents the chemical potential of the solid ice phase, μS1(T,P), is the same expression as eq 3 but written for the solid phase.37

2.2. 15

Substituting eqs 4 and 15 into eq 1 gives

2.2. 16

At the reference state, pure water is in equilibrium with pure ice; thus

2.2. 17

Substituting eq 17 into eq 16 yields

2.2. 18

which can be rearranged as follows

2.2. 19

Next, we know that the osmole fraction, Inline graphic, is a function of the fraction of ice; however, we also know that the fraction of ice present is actually a function of temperature and pressure. The last equation that must be derived is how the ice fraction changes with temperature and pressure in the system. To start, the molar ice fraction can be alternatively calculated using the molar volumes as follows42

2.2. 20

where, Inline graphic is the molar volume of the entire system (solid and liquid phases combined), Inline graphic is the molar volume of the liquid phase and is a function of temperature, pressure, fraction of ice, and initial solute concentration, and Inline graphic is the molar volume of the solid phase (pure water, Inline graphic) and is a function of temperature and pressure. In the current study, the system is a sealed constant volume (or isochoric) system; therefore, Inline graphic is constant and does not change with temperature and pressure. Therefore, the molar volume of the solution at the temperature and pressure at which the container is initially sealed (in the absence of air), Tseal and Pseal, will be the same as Inline graphic throughout the isochoric freezing. Because of this

2.2. 21

where Tseal and Pseal depend on how the experiment/protocol was conducted. Substituting eq 21 into eq 20 yields

2.2. 22

How the liquid volume, Inline graphic, depends on temperature, pressure, and solute concentration can be calculated as follows

2.2. 23

where Inline graphic is the partial molar volume of water in the solution as a function of temperature, pressure, and composition, and Inline graphic, i = 2, 3, ..., r, is the partial molar volume of solute i in solution. Equations for the partial molar volumes of species are given in Section 2.3 because they are molecular-species specific. Equation 23 can be solved at Tseal and Pseal to obtain the molar volume of the solution when first sealed

2.2. 24

Substituting eqs 23 and 24 into eq 22, gives an implicit function of ice fraction, y. Thus, eq 22 can be solved for the ice fraction at any temperature and pressure of interest, in turn making the ice fraction a function of temperature, pressure, and initial composition. As a result, eq 19 becomes purely a function of the temperature, pressure, and initial composition

2.2. 25

Equation 25 can then be solved for the freezing-point temperature at each pressure to find the equilibrium conditions for the multicomponent solution at each point during isochoric freezing. Numerical methods used for solving eq 25 are given in the Supporting Information.43 To predict the molar fraction of ice in the system, eq 22 can be solved for ice fraction at each temperature and pressure pair found when solving eq 25. However, for practical reasons, for example when designing experiments, it might be more useful to know the percentage of the system volume that is ice, yV%. The ice volume percentage can be calculated as follows

2.2. 26

Equation 9 can be rearranged for nS

2.2. 27

When eqs 10 and 27 are substituted into eq 26, ntot will cancel, and eq 26 becomes

2.2. 28

Lastly, the mole fractions of solutes in the unfrozen liquid fraction can be predicted also by solving eq 22 for the molar ice fraction at the temperatures and pressures found when solving eq 25, and then substituting these values into eq 13 for each solute, i.

2.3. System Specific Parameters and Equations

One main objective of this work is to use the method described in Section 2.2 to predict the experimental results of various authors in the literature to evaluate the performance of the method. Some parameters used to make these predictions come directly from the experimental protocols used to obtain the literature data. In this study, we predict data obtained by Rubinsky et al.,2 Beşchea et al.,9,10 and Preciado and Rubinsky.7 This section includes general parameters used (Table 1), molar masses for the species in the systems (Table 2), osmotic virial coefficients for solutes in the systems (Table 3), equations used to represent the properties of water and solutes, and finally parameters specific to each author’s method. A flowchart summarizing the steps is given in Section 2.4.

Table 1. General Parameters Used.

parameter value and units
Tref 273.15 K
Pref 101,325 Pa
R 8.314 J/molK
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Table 2. Molar Masses of Species44.

species, i molar mass, MWi [kg/mol]
H2O 0.0180153
NaCl 0.058443
glycerol 0.0920938
ethylene glycol 0.0620678
propylene glycol 0.0760944
dimethyl sulfoxide 0.078133
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Table 3. Mole-Fraction-Based Osmotic Virial Equation Parameters for Solutes32,34.

species (i) k+i B+i C+i data limita (mole fraction)
NaCl34 1.8348 0.2853 14.8930 0.0843
glycerol (GLY)32 1 3.17 0 0.227
ethylene glycol (EG)32 1 3.41 0 0.303
propylene glycol (PG)32 1 4.98 0 0.262
dimethyl sulfoxide (DMSO)32 1 2.35 43.6 0.212
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a

The data limit is the largest mole fraction of the data fit by the authors to obtain parameters.

2.3.1. Parameters for Water (H2O)

We approximate the partial molar volume of water in eq 23 with the molar volume of pure liquid water at the specified temperature and pressure

2.3.1. 29

For pure liquid water properties—Inline graphic for use in eq 23 via eq 29 and Inline graphic, Inline graphic, and Inline graphic for use in eq 25—we used the Grenke and Elliott correlation.37 We assume that the ice phase in this application is hexagonal ice (Ih) and we used the Feistel and Wagner correlation36 for ice Ih to represent Inline graphic in eq 22 and Inline graphic, Inline graphic, and Inline graphic in eq 25. Equations for each of these properties can also be found in the Supporting Information of Grenke and Elliott.37 The value used for entropy of fusion, Inline graphic, in eq 25 was obtained from the heat of fusion, Inline graphic,45 which is 6010 J/mol as follows

2.3.1. 30

2.3.2. Parameters for Sodium Chloride (NaCl)

The partial molar volume of NaCl in eq 23 is a function of temperature, pressure, and of all of the solute concentrations. We approximate the partial molar volume of NaCl as the partial molar volume of NaCl in the binary solution of water and NaCl

2.3.2. 31

To obtain an equation for Inline graphic, we started with an empirical equation for the density of a NaCl–H2O solution as a function of temperature and concentration of NaCl.46

2.3.2. 32

where

2.3.2. 33
2.3.2. 34
2.3.2. 35

where T is the temperature in K, cNaCl is the w/w% of NaCl in water, and Inline graphic is the density of the solution in kg/m3. We assume that the contribution of volume that NaCl brings to the solution is independent of pressure. Parameters for eqs 33–35 are given in Table 4.

Table 4. Parameters Used in eqs 33–35 to Calculate the Density of NaCl–H2O Solutions46.
parameter value
a11 [kg/m3] 750.2834
a12 [kg/m3] 26.7822
a13 [kg/m3] –0.26389
a21 [kg/m3K] 1.90165
a22 [kg/m3K] –0.11734
a23 [kg/m3K] 0.00175
a31 [kg/m3K2] –0.003604
a32 [kg/m3K2] 0.0001701
a33 [kg/m3K2] –0.00000261
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Equations 33–35 use concentration in w/w%; however, in this work we represent concentration in mole fraction. This conversion can be done as follows

2.3.2. 36

where MWi is the molar mass of species i in kg/mol given in Table 2, and zNaCl is the mole fraction of NaCl in water if there are no other components present in the solution. This can be obtained from the mole fraction of NaCl in the entire solution, xNaCl, as follows

2.3.2. 37

where

2.3.2. 38

In order to convert eq 32 into a partial molar volume of NaCl as appears in eq 23, it first needs to be converted into molar volume, Inline graphic, using the average molar mass, ⟨MWNaCl–H2O⟩

2.3.2. 39

where

2.3.2. 40

Next, the partial molar volume can be obtained by using the following expression47

2.3.2. 41

where Inline graphic is obtained by taking the derivative of eq 39 with respect to zNaCl, while holding temperature constant.

2.3.2. 42

where

2.3.2. 43
2.3.2. 44
2.3.2. 45
2.3.2. 46

where

2.3.2. 47

2.3.3. Parameters for Glycerol (GLY), Ethylene Glycol (EG), and Propylene Glycol (PG)

We approximate the partial molar volume of glycerol, ethylene glycol, and propylene glycol in eq 23 with the molar volume of pure glycerol, ethylene glycol, and propylene glycol, respectively, at the specified temperature and pressure

2.3.3. 48
2.3.3. 49
2.3.3. 50

To obtain Inline graphic, Inline graphic, and Inline graphic, we begin with published Tait–Tammann equations for density as a function of temperature and pressure.48

2.3.3. 51

where, ρ(T,P) is the density in kg/m3 of glycerol, ethylene glycol, or propylene glycol, Pref is atmospheric pressure (101,325 Pa), ρref(T) is the density of glycerol, ethylene glycol, or propylene glycol at Pref as a function of temperature, and T and P are temperature and pressure in K and Pa, respectively. In eq 51

2.3.3. 52
2.3.3. 53
2.3.3. 54

where

2.3.3. 55
2.3.3. 56
2.3.3. 57

where the constants shown in eqs 52–57 are given in Table 5.

Table 5. Parameters Used in eqs 51–57 to Calculate the Density of Glycerol (GLY), Ethylene Glycol (EG), and Propylene Glycol (PG)48.
parameter GLY EG PG
b1 6.9496 1.77482 1180.71
b2 –9.25236 1.11208 –0.256645
b3 5.43535 0 –0.000805917
ρc 351.51 333.7  
Tc 800 790  
b11 0.114255 0.0950140 0.0838940
b22 925.959 674.427 909.931
b33 –1.5814 –1.77594 –4.03075
b44 0.00048909 0.00127583 0.00524295
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The molar volumes of glycerol, ethylene glycol, and propylene glycol can be obtained from eq 51 as follows

2.3.3. 58
2.3.3. 59
2.3.3. 60

where MWGLY, MWEG, and MWPG are given in Table 2.

2.3.4. Parameters for Dimethyl Sulfoxide (DMSO)

We also approximate the partial molar volume of dimethyl sulfoxide with its pure species molar volume

2.3.4. 61

Within our search, we were unable to find an equation for how the density of DMSO is impacted by pressure; therefore, we assume that its density is purely a function of temperature

2.3.4. 62

We fit DMSO density data (in g/cm3) by Casteel and Sears49 to a linear expression with respect to temperature (in K) using MATLAB’s fitlm function.

2.3.4. 63

where ρDMSO is the density of DMSO in g/cm3, and values for g1 and g2 are given in Table 6. Five significant figures were required to redraw the function obtained using unrounded values from fitlm; however, to ensure accuracy, six significant figures are reported.

Table 6. Parameters Used in eq 63 to Calculate the Density of Dimethyl Sulfoxide.
parameter valuea
g1 –0.00100895
g2 1.39641
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a

These values were obtained in this work by fitting the data from Casteel and Sears.49

The molar volume (m3/mol) of dimethyl sulfoxide can be obtained from eq 63 as follows

2.3.4. 64

where MWDMSO can be found in Table 2.

2.3.5. Parameters Required to Analyze Data from Rubinsky et al.2

When Rubinsky et al.2 conducted their experiments, they sealed the isochoric chamber after nucleation, at the nucleation temperature, Tnuc; therefore, Tseal = Tnuc in eqs 21, 22, and 24. The containers were sealed at atmospheric pressure; therefore, Pseal = 101,325 Pa. The study by Rubinsky et al.2 included measurements for physiological saline, 1 and 2 M glycerol in physiological saline, and 1 and 2 M ethylene glycol in physiological saline. The initial mole fractions of solutes prior to isochoric freezing are given in Table 7, and the methods that we used to obtain these are given in the Supporting Information.

Table 7. Initial Concentrations of Solutes in Each Solution Investigated by Rubinsky et al.2a.
solution species 2 mole fraction 2 x2,initial species 3 mole fraction 3 x3,initial
physiological saline NaCl 0.002774805 N/A N/A
1 M GLY–saline NaCl 0.002720713 GLY 0.019493685
2 M GLY–saline NaCl 0.002658189 GLY 0.042026423
1 M EG–saline NaCl 0.002722383 EG 0.018892043
2 M EG–saline NaCl 0.002665682 EG 0.039326375
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a

Solutes included are NaCl (sodium chloride), GLY (glycerol), and EG (ethylene glycol). The methods that we used to obtain these values are given in the Supporting Information.

Because sealing occurred at the nucleation temperature, the nucleation temperature was calculated before making predictions. To do this, the osmole fraction for each solution was calculated using eqs 5–7, the initial mole fractions given in Table 7, and the osmotic virial coefficients given in Table 3. After obtaining the osmole fraction, Inline graphic, the nucleation temperature can be calculated as follows30

2.3.5. 65

2.3.6. Parameters Required to Analyze Data from Beşchea et al.9,10

Although not explicitly indicated in the paper, it seems as though Beşchea et al.9,10 sealed their isochoric systems at room temperature before they began cooling; therefore, we set Tseal to 298.15 K. The containers were sealed at atmospheric pressure; therefore, Pseal = 101,325 Pa. The study by Beşchea et al.9,10 included measurements for physiological saline; 1, 2, 3, 4, and 5 M glycerol in physiological saline; 1, 2, 3, 4, and 5 M ethylene glycol in physiological saline; and 1, 2, 3, and 4 M dimethyl sulfoxide in physiological saline. The initial mole fractions of solutes prior to isochoric freezing are given in Table 8, and the methods that we used to obtain these are given in the Supporting Information.

Table 8. Initial Concentrations of Solutes in Each Solution Investigated by Beşchea et al.9,10a.
solution species 2 mole fraction 2 x2,initial species 3 mole fraction 3 x3,initial
physiological saline NaCl 0.002774805 N/A N/A
1 M GLY–saline NaCl 0.002720713 GLY 0.019493685
2 M GLY–saline NaCl 0.002658189 GLY 0.042026423
3 M GLY–saline NaCl 0.002585094 GLY 0.068368975
4 M GLY–saline NaCl 0.002498498 GLY 0.099576810
5 M GLY–saline NaCl 0.002394282 GLY 0.137134937
1 M EG–saline NaCl 0.002722383 EG 0.018892043
2 M EG–saline NaCl 0.002665682 EG 0.039326375
3 M EG–saline NaCl 0.002604154 EG 0.061499895
4 M EG–saline NaCl 0.002537158 EG 0.085644508
5 M EG–saline NaCl 0.002463929 EG 0.112035267
1 M DMSO–saline NaCl 0.002721530 DMSO 0.019199320
2 M DMSO–saline NaCl 0.002661921 DMSO 0.040681719
3 M DMSO–saline NaCl 0.002594775 DMSO 0.064880174
4 M DMSO–saline NaCl 0.002518567 DMSO 0.092344542
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a

Solutes included are NaCl (sodium chloride), GLY (glycerol), EG (ethylene glycol), and DMSO (dimethyl sulfoxide). The methods that we used to obtain these values are given in the Supporting Information.

2.3.7. Parameters Required to Analyze Data from Preciado and Rubinsky7

The temperature at which Preciado and Rubinsky7 sealed their isochoric system was not explicitly indicated in their paper; however, based on the outlined protocol, it seems as though it was sealed at room temperature before cooling. Therefore, we set Tseal to 298.15 K. The containers were sealed at atmospheric pressure; therefore, Pseal = 101,325 Pa. The study by Preciado and Rubinsky7 included measurements for 1.5 M glycerol, ethylene glycol, dimethyl sulfoxide, or propylene glycol in distilled water. In this work, we only included a prediction for 1.5 M dimethyl sulfoxide in distilled water. We chose not to include the predictions for 1.5 M glycerol and ethylene glycol in distilled water, because these experimental data did not seem to agree with the experimental data of Beşchea et al.,9,10 with which our predictions agreed. Our predictions also did not agree with the data for 1.5 M propylene glycol in distilled water of Preciado and Rubinsky,7 which may mean that their propylene glycol data suffers from the same systematic difference as their glycerol and ethylene glycol data. The initial mole fraction for 1.5 M dimethyl sulfoxide in distilled water prior to isochoric freezing is given in Table 9, and the method used to obtain this value is given in the Supporting Information.

Table 9. Initial Mole Fraction of Dimethyl Sulfoxide (DMSO) in Distilled Watera.
solution species 2 mole fraction 2 x2,initial species 3 mole fraction 3 x3,initial
1.5 M DMSO–water N/A N/A DMSO 0.029455844
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a

The method used to obtain this value is given in the Supporting Information.

2.3.8. Parameters Used to Make Predictions for Solutions of Propylene Glycol, NaCl, and Water

Although, we did not find any additional experimental results for isochoric solutions containing propylene glycol, we chose to make predictions for 1, 2, 3, 4, and 5 M propylene glycol in physiological saline. This is because propylene glycol is a common cryoprotectant used in cryopreservation, and these predictions may be useful in experimental design. We chose to use a Tseal of 298.15 K and a Pseal of 101,325 Pa for this prediction. The mole fractions for each of the predicted solutions are given in Table 10, and the method used to obtain these are given in the Supporting Information.

Table 10. Initial Concentrations of Various Propylene Glycol (PG)–Sodium Chloride (NaCl)–Water Solutionsa.
solution species 2 mole fraction 2 x2,initial species 3 mole fraction 3 x3,initial
1 M PG–saline NaCl 0.002721396 PG 0.019247663
2 M PG–saline NaCl 0.002661317 PG 0.040899379
3 M PG–saline NaCl 0.002593234 PG 0.065435554
4 M PG–saline NaCl 0.002515433 PG 0.093473726
5 M PG–saline NaCl 0.002425675 PG 0.125821263
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a

The methods that we used to obtain these values are given in the Supporting Information.

2.4. Flowchart Summarizing Prediction Process

Figure 2 contains a flowchart that summarizes the theory outlined in Sections 2.2 and 2.3 to help visualize the steps used to produce the predictions.

Figure 2.

Figure 2

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General flowchart outlining prediction process given in Sections 2.2 and 2.3.

3. Results and Discussion

This section gives predictions for literature data obtained by Rubinsky et al.,2 Beşchea et al.,9,10 and Preciado and Rubinsky7 who measured the pressure increase when various solutions were cooled isochorically. Along with the pressure predictions, we also predicted the volume percentage of ice in the system at each temperature, as well as how the mole fractions of the solutes in the unfrozen fraction increase as more ice forms. Although no definitive experimental results were found in literature for the isochoric freezing of solutions containing propylene glycol, some independent predictions were made for propylene glycol in saline solutions. All predictions were made by using the method outlined in the Supporting Information to solve eq 25, using the water and solute data given in Section 2.2. As was described in Section 2, the sealing temperature for prediction of the Rubinsky et al.2 data was set to the nucleation temperature, and the sealing temperature for the prediction of the of Beşchea et al.9,10 data, Preciado and Rubinsky7 data, and for the propylene glycol solution predictions were set to room temperature (or 298.15 K). The sealing temperature impacts the predicted pressure increase, volume of ice formed, and solute concentration in the unfrozen fraction. Therefore, this section also includes an investigation of the impact of sealing temperature on isochoric freezing.

3.1. Predictions

Figures 3–11 show predictions of pressure increase (in Figures 3–10 these are compared to literature data2,7,9,10) (panel a), volume percent of ice increase (panel b), and solute concentration increase (panels c,d) as various initial concentrations of solutes are cooled isochorically. In all of the panels of Figures 3–11, the starting point should be considered the point at the furthest right (highest temperature). This point is the nucleation temperature. Although cooling begins at room temperature, 298.15 K, we do not begin making predictions until ice nucleates. Prior to this time, there is no ice formation; therefore, the concentration of solutes is constant. After nucleation, moving left on the predicted curves in Figures 3–11, indicates further isochoric freezing. In all of the panels of Figures 3–11 we see common trends. We can see in panel (a) of all Figures 3–11 that as the solution is cooled, the pressure increases. We can also see that this pressure increase is greatly reduced when higher concentrations of solutes are present. Next, in panel (b) of all Figures 3–11, we see that the volume percentage of ice increases as the temperature is lowered, and isochoric freezing occurs. This volume percentage of ice is greatly reduced when higher concentrations of solutes are used. In panels (c) and (d) of all Figures 3–11, we see that the solute concentrations increase as temperature decreases, which is understandable because there is a higher percentage of ice which concentrates the solution in the unfrozen fraction. Panel (c) in Figures 4, 5, 7–9, and 11 compares how the molar fraction of NaCl in the unfrozen fraction is impacted by the initial concentration of species 3 (GLY, EG, DMSO, or PG). We can see that when there are higher concentrations of species 3, the concentration of NaCl is lower. This is because mole fraction is calculated using the following equation (also given in of the Supporting Information)

3.1. 66

where ni is the number of moles of species i. The moles of species 3, n3, appears in eq 66. Therefore, if there is a larger concentration of species 3 (more moles of species 3), it decreases the mole fraction of NaCl. Lastly, we can see in panel (d) of Figures 4, 5, 7–9, and 11 that higher initial concentrations of species 3 lead to higher concentrations during the entire cooling process. We also see that the rate that the concentration increases is higher when there is a higher initial concentration of species 3. Lastly, whether species 3 is GLY, EG, DMSO, or PG does not impact the general trends in pressure prediction (panel a), percentage of ice (panel b), concentration of NaCl in the unfrozen fraction (panel c), and concentration of species 3 in the unfrozen fraction (panel d). However, we can still see differences between solutions containing GLY, EG, DMSO, or PG. For example, when the same concentration of GLY, EG, DMSO, and PG are used, the solutions containing DMSO have the lowest pressures, followed by those for PG, then GLY, and last those for EG. Similarly, the percentage of ice in DMSO solutions is the lowest, followed by those for PG, then GLY, and last those for EG: P(xDMSO) < P(xPG) < P(xGLY) < P(xEG); yV%(xDMSO) < yV%(xPG) < yV%(xGLY) < yV%(xEG).

Figure 3.

Figure 3

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Predictions for isochoric (constant-volume) freezing of a NaCl–H2O solution at physiological concentration (isotonic saline). (a) The pressure prediction for isotonic saline (solid red line) compared with the literature data reported by Rubinsky et al.2 (symbols). (b) The predicted volume percentage of ice for physiological saline. (c) The predicted concentration increase of NaCl in the unfrozen fraction of physiological saline.

Figure 11.

Figure 11

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Predictions for isochoric (constant-volume) freezing of NaCl–propylene glycol (PG)–H2O solutions. (a) The predicted pressure. (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of PG in the unfrozen fraction.

Figure 10.

Figure 10

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Predictions for isochoric (constant-volume) freezing of dimethyl sulfoxide (DMSO)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Preciado and Rubinsky7 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of DMSO in the unfrozen fraction.

Figure 4.

Figure 4

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Predictions for isochoric (constant-volume) freezing of NaCl–glycerol (GLY)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Rubinsky et al.2 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of GLY in the unfrozen fraction.

Figure 5.

Figure 5

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Predictions for isochoric (constant-volume) freezing of NaCl–ethylene glycol (EG)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Rubinsky et al.2 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of EG in the unfrozen fraction.

Figure 7.

Figure 7

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Predictions for isochoric (constant-volume) freezing of NaCl–glycerol (GLY)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Beşchea et al.9 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The concentration increases of GLY in the unfrozen fraction.

Figure 9.

Figure 9

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Predictions for isochoric (constant-volume) freezing of NaCl–dimethyl sulfoxide (DMSO)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Beşchea et al.10 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of DMSO in the unfrozen fraction.

Figure 6.

Figure 6

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Predictions for isochoric (constant-volume) freezing of a NaCl–H2O solution at physiological concentration (isotonic saline). (a) The pressure prediction for isotonic saline (solid red line) compared with literature data reported by Beşchea et al.9 (symbols). (b) The predicted volume percentage of ice for physiological saline. (c) The predicted concentration increase of NaCl in the unfrozen fraction of physiological saline.

Figure 8.

Figure 8

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Predictions for isochoric (constant-volume) freezing of NaCl–ethylene glycol (EG)–H2O solutions. (a) The pressure predictions (solid lines) compared with literature data reported by Beşchea et al.10 (symbols). (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of EG in the unfrozen fraction.

Table 11 gives the mean percent error (MPE) as well as the mean absolute error (MAE) between the experimental data and the solution predictions calculated using eqs 67 and 68 shown below.

3.1. 67

where N is the number of data points that were predicted.

3.1. 68

Table 11. Mean Percent Error (MPE) and Mean Absolute Error (MAE) for Each Prediction, Each Solution Combination, Each Species Combination, and Overall.

solution research group MPE/MAE
physiological saline Rubinsky et al.2 4.44%/1.31 MPa
  Beşchea et al.9 2.61%/3.03 MPa
physiological saline overall 3.30%/2.38 MPa
1 M GLY–saline Rubinsky et al.2 14.85%/1.85 MPa
  Beşchea et al.9 17.61%/12.77 MPa
  solution 16.51%/8.40 MPa
2 M GLY–saline Rubinsky et al.2 4.45%/1.59 MPa
  Beşchea et al.9 3.18%/2.27 MPa
  solution 3.77%/1.96 MPa
3 M GLY–saline Beşchea et al.9 13.17%/1.19 MPa
4 M GLY–saline Beşchea et al.9 12.75%/3.48 MPa
5 M GLY–saline Beşchea et al.9 369.05%/10.34 MPa
GLY–saline overall 27.57%/4.61 MPa
1 M EG–saline Rubinsky et al.2 35.50%/9.54 MPa
  Beşchea et al.10 7.51%/6.82 MPa
  solution 28.04%/8.81 MPa
2 M EG–saline Rubinsky et al.2 24.91%/6.82 MPa
  Beşchea et al.10 1.93%/1.69 MPa
  solution 14.86%/4.58 MPa
3 M EG–saline Beşchea et al.10 9.85%/5.15 MPa
4 M EG–saline Beşchea et al.10 11.65%/4.12 MPa
5 M EG–saline Beşchea et al.10 66.66%/9.53 MPa
EG–saline overall 22.53%/6.92 MPa
1 M DMSO–saline Beşchea et al.10 5.49%/4.92 MPa
2 M DMSO–saline Beşchea et al.10 1.50%/0.83 MPa
3 M DMSO–saline Beşchea et al.9,10 14.49%/6.02 MPa
4 M DMSO–saline Beşchea et al.9,10 10.42%/1.46 MPa
DMSO–saline overall 6.71%/3.56 MPa
1.5 M DMSO–water Preciado and Rubinsky7 40.97%/8.48 MPa
DMSO–water overall 40.97%/8.48 MPa
OVERALL 21.23%/5.44 MPa
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Note, pressure predictions were obtained by linearly interpolating between pressure predictions shown in Figures 3–10. We chose to display both the mean percent error and the mean absolute error, so that appropriate conclusions may be drawn. This is because mean percent error is sensitive to the size of P. For example, at lower pressures, the denominator in eq 67 is lower which can artificially inflate the mean percent error.

Figures 3–10 as well as Table 11 indicate some discrepancy between the experimental data and predictions. This discrepancy may in part be due to the simplified thermodynamic models for how the partial molar volume of the solutes depend on temperature, pressure, and concentration in the liquid fraction. Therefore, better thermodynamic models for the solutes of interest may lead to better predictions. Another source of error may be our assumption of the mixing temperature when converting reported experimental solution molarities to mole fraction (see the Supporting Information).

3.2. Investigating the Effects of Sealing Temperature, Tseal

To test the impact of sealing temperature on isochoric freezing predictions, we investigated solutions of isotonic NaCl (physiological saline) with 2 or 5 M glycerol. The isochoric model created in this study was used to predict the pressure increase, ice volume percent, and mole fractions of NaCl and glycerol in the unfrozen fraction, and how these differ if the isochoric chamber is sealed at the nucleation temperature (as was done by Rubinsky et al.2) or sealed at room temperature, 298.15 K (as was done by Beşchea et al.9,10). We additionally investigated a sealing temperature of 277.15 K (4 °C) because this is the temperature at which pure liquid water has the maximum density (or minimum molar volume).50Figure 12 gives these results.

Figure 12.

Figure 12

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Investigation of the effect of sealing temperature on isochoric freezing predictions for NaCl–glycerol (GLY)–H2O solutions. (a) The pressure predictions. (b) The predicted volume percentages of ice. (c) The predicted concentration increases of NaCl in the unfrozen fraction. (d) The predicted concentration increases of GLY in the unfrozen fraction.

In Figure 12a, it can be seen that if the container is sealed at 298.15 K, the final pressures are lower than when the container is sealed at the nucleation temperature or 277.15 K. We also note that this difference is greater in solutions with higher solute concentrations. Next, in Figure 12b, it can be seen that sealing at 298.15 K leads to a higher percentage of ice in the isochoric chamber than sealing at the nucleation temperature or 277.15 K; however, this difference is very minimal, leading us to speculate that sealing at 298.15 K may be a better option experimentally because it results in a pressure significantly lower while the ice fraction is marginally impacted. It can also be seen that when the container is sealed at 298.15 K, the ice fraction does not start at 0%, rather it jumps to ∼2–10%. The reason is that when the container is sealed at 298.15 K, the solution temperature is higher than the temperature where water has its lowest specific volume, 277.15 K (4 °C). If the solution were cooled under isobaric conditions, it would shrink between 298.15 and 277.15 K, and then expand during further cooling.6 However, because the solution is being cooled under isochoric (constant-volume) conditions, the pressure will instead decrease to maintain the constant volume as the solution is cooled. Thus, when the nucleation temperature is reached, there is more space for ice in the container, so at equilibrium it will take up all of the available space, ∼2–10%. This is further confirmed by the curves in Figure 12b that show the percentage of ice when the chamber is sealed at 277.15 K (the specific volume minimum). Because between this nucleation temperature and 277.15 K, the specific volume of liquid does not decrease appreciably, there would be no pressure drop, therefore, at the nucleation temperature, the fraction of ice is 0%. Lastly, in Figure 12c,d, it can be seen that the solute concentrations are slightly higher when the container is sealed at 298.15 K rather than at the nucleation temperature or 277.15 K; however, this difference is relatively insignificant. This means that if one chose to seal the container at 298.15 K in the presence of cells, it would not likely be detrimental to cell survival in terms of toxicity.

4. Conclusion

The objective of this work was to create a model that can be used to predict the pressure increase, volume percentage of ice, and the increase in solute concentrations in the unfrozen fraction during isochoric (constant-volume) freezing. We did so by combining Gibbsian thermodynamics,29,30 the Grenke and Elliott correlation for pure liquid water,37 the Feistel and Wagner correlation for ice Ih,36 and the Elliott et al. multisolute osmotic virial equation.31−35 We made predictions for systems containing physiological concentrations of NaCl, and various concentrations of glycerol, ethylene glycol, dimethyl sulfoxide, and propylene glycol. We verified the accuracy of our model by comparing our predictions to literature data reported by Rubinsky et al.,2 Beşchea et al.,9,10 and Preciado and Rubinsky.7 Our predictions had a mean percent error of 21.23% and a mean absolute error of 5.44 MPa. Additionally, we investigated how sealing the isochoric chamber at the nucleation temperature, the minimum-molar-volume temperature, or at room temperature impacts the predicted pressure increase, ice volume percentage, and unfrozen fraction solute concentrations. We found that sealing the container at room temperature significantly decreased the chamber pressure, especially at higher solute concentration. However, we found that sealing temperature had only a minor impact on ice volume percentage and solute concentration. Thus, we suggest that there may be some benefit to sealing the chamber at room temperature in order to mitigate the pressure increase. This work can further be used to aid in the design of isochoric cryopreservation experiments and protocols, and the method can be further studied in application to other cryopreservation solutes and combinations of solutes.

Acknowledgments

This research was funded by the Canadian Institutes of Health Research (CIHR PS 166125), the Natural Sciences and Engineering Research Council of Canada (NSERC), and the University of Alberta. J.A.W.E. holds a Canada Research Chair in Thermodynamics.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.4c03915.

  • (A) Numerical methods for predicting the pressure increase of multicomponent solutions during isochoric freezing, and (B) calculation of initial mole fractions of solutes (PDF)

The authors declare no competing financial interest.

Supplementary Material

jp4c03915_si_001.pdf (346.9KB, pdf)

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