Comment on ''Determination of the quaternary phase ...

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27 March 2015 Cryobiology xxx (2015) xxx–xxx 1

Contents lists available at ScienceDirect

Cryobiology journal homepage: www.elsevier.com/locate/ycryo

2

Brief Communication

8 4 9

Comment on ‘‘Determination of the quaternary phase diagram of the water–ethylene glycol–sucrose–NaCl system and a comparison between two theoretical methods for synthetic phase diagrams’’ Cryobiology 61 (2010) 52–57

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Michal W. Zielinski a,b, Locksley E. McGann b, John A. Nychka a, Janet A.W. Elliott a,b,⇑ a b

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada Department of Laboratory Medicine and Pathology, University of Alberta, Edmonton, Alberta T6G 2R8, Canada

a r t i c l e

i n f o

Article history: Received 8 November 2014 Accepted 17 March 2015 Available online xxxx Keywords: Osmotic virial Freezing point summation Thermodynamics Modeling Freezing point depression Quaternary phase diagram Ethylene glycol Sucrose NaCl

a b s t r a c t Recently, measurements of a considerable portion of the phase diagram for the quaternary system water– ethylene glycol–sucrose–NaCl were published (Han et al., 2010). In that article, the data were used to evaluate the accuracy of two non-ideal multi-solute solution theories: the Elliott et al. form of the multi-solute osmotic virial equation and the Kleinhans and Mazur freezing point summation model. Based on this evaluation, it was concluded that the freezing point summation model provides more accurate predictions for the water–ethylene glycol–sucrose–NaCl system than the multi-solute osmotic virial equation. However, this analysis suffered from a number of issues, notably including the use of inconsistent solute-specific coefficients for the multi-solute osmotic virial equation. Herein, we reanalyse the data using a recently-updated and consistent set of solute-specific coefficients (Zielinski et al., 2014). Our results indicate that the two models have very similar performance, and, in fact, the multi-solute osmotic virial equation can provide more accurate predictions than the freezing point summation model depending on the concentration units used. Ó 2015 Published by Elsevier Inc.

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Theoretical thermodynamic solution theories are a central component of many predictive models of cell and tissue behaviour during cryopreservation. A primary application of these thermodynamic models is the calculation of water chemical potential in a solution of interest, often in terms of the directly-related solution properties of osmolality and/or freezing point depression. The simplest solution theory is the ideal dilute model, where, for example, the osmolality is given as the sum of the solute concentrations. Notably, this model assumes that there are no interactions between solute molecules in solution. However, most solutions in cryobiology tend to be highly concentrated and thus thermodynamically non-ideal. As such, use of the ideal model can negatively impact prediction accuracy [4,14,18]. Alternatively, non-ideal solution theories can be used, although these models introduce additional complexity as they rely on solute-specific thermodynamic coefficients to account for solute–solute interactions. Such coefficients are typically obtained empirically, i.e. by curve-fitting to experimental solution data [3,4,9,14,18]. Cryobiological ⇑ Corresponding author at: Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada. Fax: +1 780 492 2331. E-mail address: [email protected] (J.A.W. Elliott).

solutions are predominantly multi-solute in nature. In order to model a solution containing multiple solutes, non-ideal solution theories can often require curve-fitting to solution data measured in the presence of all the solutes of interest, effectively rendering the obtained coefficients solution-specific instead of solute-specific. Indeed, many of the non-ideal multi-solute solution theories that have been proposed for use with cryobiological solutions require fitting to multi-solute solution data in this way, limiting their applicability to the specific solutions to which they have been fit [1,5,11–13,17]. Given the wide variety of combinations of many different types of solutes that cryobiological solutions can contain, and the difficulties of measuring multi-solute solution data [9], these types of non-ideal models do not represent an approach that can be applied to cryobiological solutions in general. In contrast to these solution-specific models, at least two solution theories have been proposed that rely only on single-solute solution data to make predictions in multi-solute solutions—namely, the Elliott et al. form of the multi-solute osmotic virial equation [4] and the Kleinhans and Mazur freezing point summation model [9]. Experimental multi-solute phase diagrams provide a means of evaluating the accuracy of theoretical multi-solute solution theories, as the model predictions can be compared to the experimental

http://dx.doi.org/10.1016/j.cryobiol.2015.03.006 0011-2240/Ó 2015 Published by Elsevier Inc.

Please cite this article in press as: M.W. Zielinski et al., Comment on ‘‘Determination of the quaternary phase diagram of the water–ethylene glycol– sucrose–NaCl system and a comparison between two theoretical methods for synthetic phase diagrams’’ Cryobiology 61 (2010) 52–57, Cryobiology (2015), http://dx.doi.org/10.1016/j.cryobiol.2015.03.006

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data [1,4,7,14,18]. Recently, Han et al. [7] published an extensive set of freezing point depression data for the quaternary system water–ethylene glycol–sucrose–NaCl. In that work, the experimental data were also used to evaluate the prediction accuracy of the Elliott et al. and the Kleinhans and Mazur models. Based on their analysis, Han et al. asserted that, for the water–ethylene glycol– sucrose–NaCl system, the Kleinhans and Mazur freezing point summation model provides more accurate predictions of freezing point depression than does the Elliott et al. multi-solute osmotic virial equation. However, in their analysis, although Han et al. implemented the freezing point summation model exactly as described by Kleinhans and Mazur [9], the multi-solute osmotic virial equation was not used in a way consistent with the description provided by Elliott et al. [4]. Specifically, the osmotic virial coefficients used by Han et al., although cited as being obtained from Elliott et al.’s original presentation [4] of their model, could not have come from that work, as Elliott et al. did not provide osmotic virial coefficients for ethylene glycol, sucrose, or NaCl. It is possible that the coefficients used came from a later work by the same group [14], but the dissociation constant for NaCl used by Han et al. is not consistent with that work, and appears to come from a different source altogether [8]. In addition, although the Elliott et al. multi-solute osmotic virial equation was proposed in two forms expressed in terms of either molality or mole fraction, only the molality-based form was considered in Han et al.’s analysis. We have recently shown [18] that although these two forms of the multi-solute osmotic virial equation appear very similar, their performance can vary considerably for a given solution system. Finally, it should be noted that while Han et al. compared the predictions of the two non-ideal models to their experimental measurements, no comparison was made to predictions made using an ideal dilute model. Ideal model predictions provide a baseline reference which can be used to help judge the relative improvement in prediction accuracy provided by a given non-ideal model, as compared to alternative non-ideal models. In this work, we revisit the analysis of Han et al., while addressing the points made above. Specifically, we make predictions of freezing point depression in the water–ethylene glycol–sucrose– NaCl system using the Kleinhans and Mazur freezing point summation model, the molality- and mole fraction-based forms of the Elliott et al. multi-solute osmotic virial equation, and an ideal dilute model, and then compare these predictions to Han et al.’s experimental measurements to analyze each model’s accuracy. Both non-ideal solution theories express water chemical potential (i.e. osmolality or freezing point depression) as a polynomial in terms of solute concentration. Briefly, for a solution containing (r 1) solutes, the Kleinhans and Mazur freezing point summation model is defined as [9]

DT m ¼ T om T m ¼

r X i¼2

C 1i mi þ C 2i m2i þ C 3i m3i ;

ð1Þ

where DTm is the freezing point depression of the solution (K), Tm is the solution’s absolute freezing point (K), Tm is the absolute freezing point of pure water (K), mi is the molality of solute i (moles solute/ kg water), and C1i, C2i, and C3i are empirical solute-specific coefficients for solute i (°C [moles solute/kg water]1, °C [moles solute/ kg water]2, and °C [moles solute/kg water]3, respectively). For that same solution, the molality- and mole fraction-based forms of the Elliott et al. multi-solute osmotic virial equation are, respectively, defined as [4,14]

r r X r X X ðBii þ Bjj Þ ki mi kj mj p ¼ ki mi þ 2 i¼2 i¼2 j¼2 þ 150

r X r X r h X i¼2 j¼2 k¼2

i ðC iii C jjj C kkk Þ1=3 ki mi kj mj kk mk þ . . . ;

ð2Þ

r r X r X X ðBii þ Bjj Þ p~ ¼ ki xi þ ki xi kj xj 2 i¼2 i¼2 j¼2 þ

r X r X r h X

ðC iii C jjj C kkk Þ

1=3 ki xi kj xj kk xk

i¼2 j¼2 k¼2

i

þ ...;

ð3Þ 153

~ is the where p is the osmolality of the solution (osmoles/kg), p related but less-commonly used osmole fraction of the solution (dimensionless), xi is the mole fraction of solute i (dimensionless), Bii and C iii are the molality-based second and third osmotic virial coefficients of solute i ([moles solute/kg water]1 and [moles solute/ kg water]2, respectively), Bii and C iii are the mole fraction-based second and third osmotic virial coefficients of solute i (dimension less), and ki and ki are, respectively, the molality- and mole fraction-based dissociation constants of solute i (dimensionless). The dissociation constant ki (or ki ) is required for electrolyte solutes, where it accounts for complexities such as ionic dissociation and charge screening [15]; for any non-electrolyte solute its value is 1. Osmolality and osmole fraction are related by [4].

p~ ¼ M1 p;

ð4Þ

where M1 is the molar mass of water (kg/mol). Osmolality can be converted to freezing point depression using [14]

RT om

½M1 =Dsof1

p DT m ¼ ; 1 þ Rp½M1 =Dsof1

154 155 156 157 158 159 160 161 162 163 164 165 166

167 169 170 171

172

ð5Þ 174

where R is the universal gas constant and

Dsof1

is the standard molar

175

entropy change of fusion of water. The values and units of M 1 , R, T om ,

176

and Dsof1 can be found in Table 1.

177

A consistent set of solute-specific coefficients for the freezing point summation model and for both forms of the multi-solute osmotic virial equation has recently been obtained for a number of solutes of cryobiological interest [18]. The coefficients for ethylene glycol, sucrose, and NaCl from that work were used in all calculations herein. Note that these coefficients were obtained using single-solute data available in the literature for ethylene glycol [10,16], for sucrose [6,16], and for NaCl [16]. Table 2 contains a summary of these coefficients. In addition to the above non-ideal models, a molality-based ideal dilute model was considered in this work, specifically,

178

r X p ¼ ki mi :

ð6Þ

i¼2

The raw experimental data provided by Han et al. [7] (as online supplementary data) is in the form of concentration in mass fraction, and, more specifically, is organized into isopleths in terms of the following solute mass ratios

W EG Re ¼ ; W NaCl þ W Suc

ð7Þ

179 180 181 182 183 184 185 186 187 188

189

191 192 193 194 195

196 198 199

W Suc Rs ¼ ; W NaCl

ð8Þ

and the total solute mass fraction W t ¼ W NaCl þ W Suc þ W EG , where W NaCl is the mass fraction of NaCl, W Suc is the mass fraction of

Table 1 Values and units of constants used in Eqs. (4) and (5) [2]. Constant

Value

R M1 T om

8.314 J/(mol K) = 8.314 Pa m3/(mol K) 1.802 102 kg/mol 273.15 K 22.00 J/(mol K)

Dsof1


201 202 203

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Table 2 Elliott et al. molality- and mole fraction-based osmotic virial coefficients, Kleinhans and Mazur freezing point summation model coefficients, and molar masses of NaCl, sucrose, and ethylene glycol (EG).a

Kleinhans and Mazur freezing point summation model coefficients [18]

C1i (°C/molal) C2i (°C/molal2) C3i (°C/molal3) ki Bii (molal1) Ciii (molal2) ki Bii Ciii

Elliott et al. molality-based osmotic virial coefficients [18]

Elliott et al. mole fraction-based osmotic virial coefficients [18]

Molar mass (kg/mol) [16]

NaCl

Sucrose

EG

3.357 0.0043 2.56 102 1.678 0.044 0 1.644 3.80 0 5.844 102

1.824 0.2825 1.84 102 1 0.116 0 1 8.68 0 3.423 101

1.814 0.0548 1.76 103 1 0.020 0 1 3.41 0 6.207 102

a

In the course of preparing this manuscript, we uncovered a minor transcription error in our previous article [18] which had no impact on the analysis in that work. Specifically, the maximum considered solute concentration listed for ethylene glycol (EG) in Table 5 of [18] for the Kleinhans and Mazur freezing point summation model, 9.062 mol/kg, is incorrect. The correct value—and the one that was used in all calculations—is 24.166 mol/kg, as listed in Table 4 of [18] for the Elliott et al. molality-based multi-solute osmotic virial equation. 204 205 206 207 208 209

210

212

sucrose, and W EG is the mass fraction of ethylene glycol. Each isopleth consists of 10–12 data points of increasing W t at constant Re and Rs . As the solution models being considered here (i.e. Eqs. (1)–(3) and (6)) express concentration in terms of either molality or mole fraction, the following conversion equations were required to make model predictions of Han et al.’s measurements

W NaCl ¼

Wt ; ðRe þ 1ÞðRs þ 1Þ

ð9Þ

213 215

W Suc ¼

Rs W t ; ðRe þ 1ÞðRs þ 1Þ

ð10Þ

Re W t ; ðRe þ 1Þ

ð11Þ

216 218

W EG ¼

219 221

mi ¼

Wi ; Mi W 1

ð12Þ

222

xi ¼ 224 225 226 227 228 229 230 231

232

234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250

Wi M P iW i i Mi

¼W

Wi Mi NaCl

M NaCl

Suc EG 1 þW þW þW M Suc M EG M1

;

ð13Þ

where W 1 ¼ 1 W t is the mass fraction of water and Mi is the molar mass of species i. The molar masses of NaCl, sucrose, and ethylene glycol are given in Table 2. The accuracy of model predictions was evaluated using the same measure employed by Han et al. [7]—the standard deviation of the model predictions from the experimental measurements (r), calculated on a per-isopleth basis, i.e.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN ex pr 2 a¼1 ðDT m;a DT m;a Þ r¼ ; N1 where

DT ex m;a

ð14Þ

is the experimentally-measured freezing point depres-

sion for data point a, DT pr m;a is the model-predicted freezing point depression for data point a, and N is the total number of data points in the isopleth being considered. The mean of the standard deviations over all the experimental isopleths was also calculated for each model. The results of the analysis for all four models are contained in Fig. 1. The isopleth-specific data in Fig. 1 is graded by colour according to the magnitude of the corresponding model’s standard deviation for that isopleth (see scale at right), with the intent of helping to reveal trends in the accuracy of each model’s predictions as a function of the relative amounts of sucrose (Rs ) and ethylene glycol (Re ). Although there do not appear to be any such trends in the prediction accuracy of any of the models, a few points do stand out from those around them in all three of the non-ideal models, including (Rs ¼ 4, Re ¼ 2) and (Rs ¼ 9, Re ¼ 2). These points

are of note because in the raw data provided by Han et al. [7], the freezing point values for these two points are identical to those of two other points: (Rs ¼ 4, Re ¼ 1) and (Rs ¼ 9, Re ¼ 3), respectively. These repeats may be the result of an error, which could help explain the discrepancies between (Rs ¼ 4, Re ¼ 2) and (Rs ¼ 9, Re ¼ 2) and the points around them. The overall mean standard deviations for all four models were recalculated while excluding all four potentially duplicated isopleths, but these exclusions appeared to have minimal effect on the results (see Fig. 1). When Han et al. originally analyzed this data and found that the Kleinhans and Mazur freezing point summation model provided more accurate predictions than the Elliott et al. molality-based multi-solute osmotic virial equation, the overall mean standard deviations that they calculated were 0.96 K for the freezing point summation model and 1.46 K for the multi-solute osmotic virial equation [7]. Here, the results are much closer: 1.02 K and 1.09 K, respectively (1.00 K and 1.08 K with potential duplicates excluded), indicating that these two models in fact have very similar performance in this solution system.1 Moreover, the mole fraction-based form of the multi-solute osmotic virial equation provides the most accurate predictions with the lowest overall mean standard deviation of 0.57 K (0.55 K with potential duplicates excluded). It is, however, important to put these results into context: the ideal model considered here has an overall mean standard deviation (5.36 K, or 5.39 K with potential duplicates excluded) that is nearly five times that of the worst-performing of the non-ideal models. In light of this result, the relative differences between the three non-ideal models are comparatively minor. Nonetheless, if the most accurate possible predictions in the water–ethylene glycol–sucrose–NaCl solution system are desired, the mole fractionbased multi-solute osmotic virial equation should be used. These results are consistent with those from our earlier work [18], where a similar analysis was conducted using the same data and solutespecific coefficients, but different measures of model accuracy. Although most aqueous solutions are highly concentrated and thus non-ideal at the low temperatures characteristic of cryobiology, there are some situations relevant to cryobiology where the solutions are dilute or near-dilute (e.g. addition/removal of low concentrations of cryoprotectant at temperatures above freezing). To examine the effect of model choice under more

1 It should be noted that when we used the original solute-specific coefficients provided by Kleinhans and Mazur [9] (as Han et al. had done in their work [7]) instead of the updated values in Table 2, we obtained the same value of the overall mean standard deviation for the freezing point summation model as Han et al. had (i.e. 0.96 K). In fact, with three exceptions, including the two points discussed above ([Rs ¼ 4, Re ¼ 2] and [Rs ¼ 9, Re ¼ 2]), we obtained identical values of standard deviation for every isopleth in the data set. The third exception is (Rs ¼ 4, Re ¼ 4); we cannot explain why our result for this isopleth differs from that of Han et al.


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Ideal Dilute Model

Rs

Re

1 2 3 4 5 6 7 8 9 10

1 6.05 5.68 5.62 6.02 5.59 5.91 5.56 5.79 6.16 6.03

2 4.56 5.57 5.20 5.58 5.64 5.49 5.51 5.48 5.73 5.57

3 4.87 5.17 5.48 5.06 5.34 5.09 5.24 5.59 5.16 5.45

4 4.37 3.01 5.65 5.51 5.21 5.63 5.58 6.08 5.92 5.33

5 4.33 5.34 5.13 5.12 5.59 6.25 5.40 6.10 6.12 6.14

1 2 3 4 5 6 7 8 9 10

1 2.44 2.13 1.01 1.35 0.78 1.07 0.60 0.69 1.11 1.00

2 1.02 1.56 0.90 1.12 1.09 0.71 0.78 0.64 0.82 0.64

3 1.75 1.34 1.34 0.71 0.78 0.63 0.62 0.88 0.51 0.71

4 1.45 0.90 1.46 1.15 0.62 1.01 0.87 1.25 1.08 0.54

5 1.57 1.67 1.11 0.93 1.32 1.54 0.82 1.48 1.51 1.41

1 2 3 4 5 6 7 8 9 10

1 0.87 0.93 1.25 1.25 0.92 1.11 0.79 0.83 1.30 1.23

2 1.79 1.02 0.87 1.00 1.10 0.84 1.01 0.90 1.12 1.00

3 0.99 0.76 1.02 0.76 0.88 0.70 0.73 1.16 0.83 1.09

4 0.84 1.94 0.99 1.05 0.72 1.16 1.01 1.53 1.46 0.84

5 0.82 0.88 0.88 0.97 1.48 1.69 1.13 1.85 1.94 1.84

1 2 3 4 5 6 7 8 9 10

1 0.63 0.56 0.82 0.58 0.52 0.46 0.35 0.46 0.57 0.54

2 1.48 0.57 0.58 0.44 0.47 0.19 0.37 0.27 0.40 0.30

3 0.67 0.36 0.43 0.47 0.35 0.75 0.60 0.48 0.39 0.39

4 0.76 2.23 0.60 0.41 0.23 0.50 0.72 0.85 0.69 0.48

5 0.65 0.52 0.38 0.44 0.80 1.05 0.37 1.04 1.12 1.04

6 7 8 4.89 4.76 5.05 4.47 4.54 4.41 4.84 4.86 4.66 5.35 5.21 5.17 5.57 4.61 5.73 5.46 5.27 5.45 5.04 5.60 5.08 5.67 5.08 5.19 5.26 5.18 5.31 5.92 5.13 5.11 Overall Mean:

9 10 4.70 5.21 5.82 5.33 4.91 5.21 4.95 5.11 5.58 5.70 5.43 5.58 5.61 5.64 5.68 5.75 5.48 5.94 5.22 5.84 5.36 (5.39)

Kleinhans and Mazur Freezing Point SummaƟon Model

Rs

Re

6 7 8 1.27 1.17 1.48 0.96 1.21 0.90 0.80 0.97 0.74 1.07 0.96 0.99 1.15 0.32 1.35 1.00 0.64 0.93 0.35 0.91 0.55 0.92 0.51 0.56 0.73 0.46 0.49 0.99 0.41 0.49 Overall Mean:

9 10 1.22 1.68 2.34 1.22 0.97 1.26 0.75 1.03 1.21 1.21 0.88 1.08 0.99 1.21 1.02 0.93 0.75 1.10 0.46 1.04 1.02 (1.00)

EllioƩ et al. Molality-Based MulƟ-Solute OsmoƟc Virial EquaƟon

Rs

Re

6 7 8 1.37 1.21 0.84 0.70 0.96 0.51 0.63 0.84 0.68 1.03 0.98 1.04 1.27 0.55 1.50 1.23 0.92 1.20 0.67 1.24 0.90 1.30 0.90 0.93 1.08 0.86 0.93 1.47 0.80 0.81 Overall Mean:

9 10 1.03 0.52 1.75 1.41 0.83 1.09 0.83 1.14 1.38 1.34 1.16 1.29 1.36 1.57 1.43 1.32 1.18 1.51 0.86 1.53 1.09 (1.08)

EllioƩ et al. Mole FracƟon-Based MulƟ-Solute OsmoƟc Virial EquaƟon

Rs

Re

6 7 8 0.99 0.94 0.54 0.53 0.66 0.50 0.30 0.39 0.47 0.45 0.29 0.38 0.55 0.54 0.75 0.53 0.24 0.45 0.31 0.47 0.32 0.52 0.26 0.35 0.46 0.27 0.31 0.69 0.37 0.48 Overall Mean:

9 10 0.89 0.37 1.17 0.87 0.29 0.40 0.24 0.45 0.63 0.69 0.39 0.72 0.55 0.78 0.62 0.63 0.54 0.83 0.33 0.76 0.57 (0.55)

Standard DeviaƟon Colour Scale 6.0+ 5.8 5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Fig. 1. Standard deviations of each model’s predictions from Han et al.’s experimental data. Each colour-graded numerical entry represents the standard deviation of the corresponding model’s predictions for a given isopleth (i.e. unique combination of Re and Rs). For each model, the mean standard deviation over all the isopleths is located immediately below the isopleth-specific values. The value outside of the brackets represents the overall mean calculated using the data from all of the isopleths. The value in brackets represents the overall mean calculated without the data points suspected of being duplicates, i.e. (Rs ¼ 4, Re ¼ 2) and (Rs ¼ 4, Re ¼ 1), and (Rs ¼ 9, Re ¼ 3), and (Rs ¼ 9, Re ¼ 2). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

291 292 293 294 295 296 297

thermodynamically ideal conditions, the above analysis was repeated while considering only experimental data points for which W t 6 0:2 (the original data of Han et al. goes up to W t ¼ 0:6 in some cases). The results of this second analysis are contained in Fig. 2. Overall, these results largely mirror those obtained using all of the available data, although, as one might expect, the differences in the performance of the four models are not as pronounced.

The extensive data set measured by Han et al. for freezing point depression in the quaternary system water–ethylene glycol– sucrose–NaCl provides a valuable resource for measuring the accuracy of non-ideal solution models to be used in cryobiology. However, in contrast to the analysis of Han et al., our investigation shows minimal difference between the prediction accuracy of the molality-based multi-solute osmotic virial equation and the


298 299 300 301 302 303 304

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Ideal Dilute Model

Rs

Re

1 2 3 4 5 6 7 8 9 10

1 1.49 0.61 0.69 1.09 0.75 0.94 0.71 0.62 0.87 0.84

2 0.38 1.09 0.79 1.08 1.02 0.88 0.63 0.93 0.81 0.72

3 0.64 0.66 0.86 0.61 0.67 0.42 0.67 0.74 0.74 0.78

4 0.66 0.23 0.86 0.80 0.57 0.98 0.77 1.00 1.08 0.43

5 0.81 0.80 0.84 0.75 0.48 0.84 0.84 1.61 0.74 1.36

1 2 3 4 5 6 7 8 9 10

1 1.09 0.44 0.49 0.67 0.49 0.54 0.33 0.51 0.47 0.40

2 0.26 0.78 0.42 0.72 0.62 0.46 0.36 0.52 0.43 0.41

3 0.40 0.36 0.55 0.32 0.39 0.47 0.34 0.33 0.39 0.35

4 0.47 0.22 0.69 0.47 0.29 0.60 0.39 0.58 0.69 0.38

5 0.61 0.55 0.56 0.46 0.46 0.55 0.46 1.20 0.35 0.94

6 7 8 0.53 0.43 0.35 0.69 0.96 0.51 0.49 0.73 0.53 0.71 0.79 0.92 1.05 0.52 1.10 0.80 0.73 0.62 0.60 0.80 0.79 0.56 0.65 0.72 0.64 0.59 0.63 0.78 0.68 0.58 Overall Mean:

9 10 0.56 0.52 1.11 0.83 0.86 0.67 0.64 0.75 1.05 0.81 0.80 0.74 1.03 0.84 1.02 0.77 0.71 1.03 0.82 1.24 0.77 (0.78)

Kleinhans and Mazur Freezing Point SummaƟon Model

Rs

Re

6 7 8 0.41 0.27 0.17 0.43 0.74 0.29 0.27 0.43 0.49 0.42 0.46 0.64 0.74 0.17 0.74 0.44 0.37 0.26 0.33 0.41 0.48 0.20 0.26 0.50 0.25 0.27 0.39 0.42 0.39 0.41 Overall Mean:

9 10 0.40 0.35 0.86 0.60 0.59 0.37 0.33 0.43 0.72 0.45 0.44 0.40 0.66 0.51 0.64 0.38 0.36 0.63 0.40 0.84 0.48 (0.47)

EllioƩ et al. Molality-Based MulƟ-Solute OsmoƟc Virial EquaƟon

Re

1 0.88 0.45 0.48 0.63 0.49 0.53 0.33 0.52 0.50 0.44

2 0.34 0.69 0.36 0.69 0.61 0.47 0.36 0.55 0.47 0.44

3 0.27 0.27 0.50 0.32 0.40 0.47 0.37 0.38 0.44 0.41

4 0.36 0.28 0.69 0.45 0.30 0.62 0.42 0.63 0.73 0.37

5 0.48 0.48 0.53 0.47 0.48 0.57 0.49 1.25 0.39 1.00

6 7 8 0.38 0.20 0.07 0.36 0.67 0.24 0.27 0.40 0.52 0.42 0.46 0.63 0.74 0.20 0.75 0.47 0.39 0.30 0.37 0.45 0.53 0.23 0.31 0.53 0.29 0.31 0.43 0.47 0.41 0.43 Overall Mean:

3.0+ 2.8 2.6 2.4 2.2 2.0

Rs 1 2 3 4 5 6 7 8 9 10

Standard DeviaƟon Colour Scale

9 10 0.29 0.23 0.80 0.53 0.58 0.35 0.34 0.43 0.72 0.47 0.46 0.41 0.69 0.53 0.68 0.43 0.40 0.68 0.47 0.90 0.48 (0.48)

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

EllioƩ et al. Mole FracƟon-Based MulƟ-Solute OsmoƟc Virial EquaƟon

Rs

Re

1 2 3 4 5 6 7 8 9 10

1 0.73 0.56 0.56 0.39 0.51 0.33 0.24 0.65 0.34 0.20

2 0.45 0.52 0.13 0.48 0.37 0.22 0.35 0.31 0.34 0.40

3 0.18 0.13 0.33 0.31 0.37 0.66 0.31 0.20 0.35 0.19

4 0.29 0.45 0.68 0.27 0.32 0.39 0.20 0.36 0.48 0.55

5 0.36 0.34 0.40 0.37 0.63 0.48 0.26 0.98 0.17 0.72

6 7 8 0.38 0.20 0.12 0.19 0.53 0.20 0.32 0.20 0.63 0.34 0.29 0.49 0.56 0.17 0.49 0.29 0.18 0.19 0.38 0.24 0.44 0.18 0.12 0.53 0.11 0.30 0.46 0.33 0.35 0.50 Overall Mean:

9 10 0.20 0.11 0.64 0.37 0.45 0.20 0.25 0.25 0.50 0.25 0.23 0.22 0.45 0.34 0.43 0.20 0.23 0.41 0.20 0.63 0.36 (0.36)

Fig. 2. Standard deviations of each model’s predictions from Han et al.’s experimental data, restricted to lower concentrations. Specifically, only data points with W t 6 0:2 were considered. As for Fig. 1, each colour-graded entry corresponds to a single isopleth, overall mean standard deviations are located below the isopleth-specific values for each model, and the overall mean values in brackets are recalculated without suspected duplicates. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

310

freezing point summation model. Moreover, what little difference exists is further marginalized when taking into consideration the far less accurate predictions provided by an ideal solution model. Finally, we show that the most accurate predictions can be achieved using the mole fraction-based form of the multi-solute osmotic virial equation.

311

Acknowledgments

312

This work was funded by the Canadian Institutes of Health Research (CIHR) (MOP 86492, OGBF INO 126778, and INO 131572), the Natural Sciences and Engineering Research Council

305 306 307 308 309

313 314

(NSERC) of Canada, the University of Alberta, and Alberta Innovates - Technology Futures. J.A.W. Elliott holds a Canada Research Chair in Thermodynamics.

315

References

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