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Osmotic Pressure beyond Concentration Restrictions Alessandro Grattoni† and Manuele Merlo‡ Dipartimento di Meccanica, Politecnico di Torino, C.so Duca degli Abruzzi, 24, 10129 Torino, Italy
Mauro Ferrari* Brown Foundation Institute of Molecular Medicine, Department of Biomedical Engineering, The UniVersity of Texas Health Science Center at Houston, Suite 537, 1825 Pressler Street, Houston, Texas 77030, Department of Experimental Therapeutics, The UniVersity of Texas M. D. Anderson Cancer Center, Houston, Texas, and Department of Bioengineering, Rice UniVersity , Houston, Texas ReceiVed: July 24, 2007
Osmosis is a fundamental physical process that involves the transit of solvent molecules across a membrane separating two liquid solutions. Osmosis plays a role in many biological processes such as fluid exchange in animal cells (Cell Biochem. Biophys. 2005, 42, 277-345;1 J. Periodontol. 2007, 78, 757-7632) and water transport in plants. It is also involved in many technological applications such as drug delivery systems (Crit. ReV. Ther. Drug. 2004, 21, 477-520;3 J. Micro-Electromech. Syst. 2004, 13, 75-824) and water purification. Extensive attention has been dedicated in the past to the modeling of osmosis, starting with the classical theories of van’t Hoff and Morse. These are predictive, in the sense that they do not involve adjustable parameters; however, they are directly applicable only to limited regimes of dilute solute concentrations. Extensions beyond the domains of validity of these classical theories have required recourse to fitting parameters, transitioning therefore to semiempirical, or nonpredictive models. A novel approach was presented by Granik et al., which is not a priori restricted in concentration domains, presents no adjustable parameters, and is mechanistic, in the sense that it is based on a coupled diffusion model. In this work, we examine the validity of predictive theories of osmosis, by comparison with our new experimental results, and a metaanalysis of literature data.
Introduction Osmosis has been studied intensely for over a hundred years and several theories were developed before a mechanistic approach was attempted. The best known osmotic pressure theories, applicable to ideal diluted mixtures, are the van’t Hoff equation5
π ) RTC2
(1)
π ) RTm′2
(2)
and the Morse equation6
where π is the osmotic pressure, R is the gas constant, T is the absolute temperature. C2 and m′2 are the solute molar and volume molal concentrations, respectively, and m′2 is defined as
m′2 ) m2F°1
(3)
where m2 is the solute molal concentration and F°1 is the density of a pure solvent at temperature T and pressure P. The van’t Hoff theory (1) has been known to fail even for very * Corresponding author. Tel: (713)-500-2444. Fax: (713)-500-2462. E-mail:
[email protected]. † Current address: Brown Foundation Institute of Molecular Medicine, Department of Nanomedicine, The University of Texas Health Science Center at Houston, 1825 Pressler St., Houston, TX, 77030. E-mail:
[email protected]. ‡ E-mail:
[email protected].
diluted solutions of several molecules including hemoglobin,8 poly(ethylene glycol),9 and BSA.10 Nonetheless, when the concentration of these molecules is sufficiently low, their solutions behave ideally. The Morse equation (2) is subject to similar limitations, but neither it, nor the van’t Hoff law (1), are restricted a priori by limits of C2, per se. The applicability of either theory to each given set of conditions must be determined empirically, and therefore these theories (and indeed all the semiempirical theories with adjustable parameters11-14) are “nonpredictive”. In recent years, novel theoretical15,16 and simulation17 models have been developed, with remarkable improvements in the study of the osmotic phenomena, but they can only be applied to limited ranges of conditions. At the same time, an interesting mechanistic approach18-20 has appeared, focused on membrane transport processes. Notwithstanding these constraints, theses theories are applicable in many cases of practical importance, such as the analytical determination of molecular weights of solute macromolecules,21 the determination of the equivalent radii of membrane pores and channels,22 facilitating molecular recognition,23 probing the behavior of biological macromolecules.24,25 A Novel Theory Granik et al.7 proposed a new osmotic pressure theory based on a mechanistic approach. The theory, summarized below, provides a simple technique for predicting osmotic pressure. In its most general formulation, it treats incompressible, nonelec-
10.1021/jp075834j CCC: $37.00 © 2007 American Chemical Society Published on Web 09/19/2007
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J. Phys. Chem. B, Vol. 111, No. 40, 2007 11771
JD ) DAΨh-1
(9)
where
Ψ) C1,A + λAτ2,AC2,A - (C1,B + λBτ2,BC2,B)(C1,A + λAC2,A) (C1,B + λBC2,B) (10)
Figure 1. Representation of osmotic pressure π in terms of two solutions separated by a membrane with two uncoupled diffusion fluxes JAfB and JBfA.
trolytic binary solutions, separated by a semipermeable or leaky membrane, at concentrations beyond the dilute, ideal regime. The theory requires no adjustable parameters and imposes no restrictions for the solution concentration or ideality, thereby offering a “predictive” capacity. The theory is based on the diffusion hypothesis: osmosis is considered as a result of spontaneous diffusion processes, in the most general case, of both solute and solvent molecules. Mechanistic equations of osmotic pressure, based on irreversible thermodynamics cannot be mathematically derived when a set of n g 2 coupled diffusion fluxes are considered. Osmosis was then regarded as a set of independent uncoupled diffusion flows of solutions. In the modeling approach that was employed for this, the diffusion processes of solvent and solute molecules were considered together as a single solution diffusion flow, and the opposite solution flows across the membrane were considered as uncoupled diffusion fluxes. It was assumed that the concept of diffusion coefficient for the solution could be introduced, beyond the conventional limitation to single species. Two uncoupled diffusion fluxes JAfB and JBfA (mol × m-2 × s-1) were then considered, where JAfB (JBfA) is the flux of a liquid binary solution A (B) toward a liquid binary solution B (A) across a rigid membrane M (Figure 1). These fluxes are described by Fick’s laws.
Cn,i are the molar concentrations of the solvent (n ) 1) and the solute (n ) 2) of the ith solutions A and B, λi are dimensionless parameters that provide a measure of particle size similarity between solute and solvent, expressed as the ratio between their molar volumes. τ2,A and τ2,B represent the fraction of solute molecules passing through the membrane from chambers A and B, respectively. A second expression of JD is obtained by considering it as a homogeneous flux of solvent and solute molecules of solution A toward compartment B, across M:
JD ) 〈CA〉VA
(11)
where 〈CA〉 is constant and is the average concentration of JD in M and VA is the flux velocity. It is expressed as
V A ) b AX A
(12)
where bA is the mechanical mobility of the particles across M and XA is the force driving the particle through the membrane against the counterflow JBfA. The driving force acting on the membrane volume VM, directed from compartment A to compartment B can be expressed as F ) N〈C〉VMXA ) N〈C〉AhXA, where N is Avogadro’s number and A is the membrane cross section area. By dividing the force F by A, an expression of stress is obtained, which is, by definition, equal to the osmotic pressure π
π ) Nh〈C〉XA
(13)
By equating the JD expressions 9 and 11 and considering eq 13, the osmotic pressure expression becomes
π ) NDAΨbA-1
(14)
JAfB ) -DA grad CA
(4)
JBfA ) -DB grad CB
(5)
Equation 14 can be modified with Einstein formula DA ) kTbA, where k ) R/N is the Boltzmann constant and the most general form of the osmotic pressure is finally derived
∂CA ) DA div grad CA ∂t
(6)
π ) RTΨ
(7)
For the special case of a pure solvent A and binary solution B separated by a semipermeable membrane, the general formulation of the theory, in the molal scale, reduces to
∂CB ) DB div grad CB ∂t
where CA, CB, DA, and DB are the variable molar concentrations and the diffusion coefficients of the diffusing species A and B, respectively. The problem is simplified by assuming the opposite fluxes directed on the x-axes, and diffusion as a stationary process that ensues on both sides of membrane M, when concentrations CA and CB are fixed. The exact solution of eqs 4-7, expressed as net diffusion flux JD is thus obtained:
JD ) DA[〈CA〉 + ξ〈CB〉]h-1
(8)
where ξ ) DB/DA and h is the thickness of the membrane. Further algebraic manipulations result in the flux expression
π)
RTm′2V [1 + (1 + V)m′2/m′1]
(15)
(16)
where V represents the volume of a solution of m2 moles of solute dissolved in 1 L of solvent. Volume V is strongly related to the volume of the solute in solution and can be expressed as V ) 1 + V2m2, where V2 is the molal volume of the dissolved solute. Equation 16 indicates that the osmotic pressure is not solely a function of the number of solute particles in solution, but it is also related to the solute size. This represents a departure from, or rather an extension of, the conventional understanding of osmosis. The notion that the solute concentration alone is
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Figure 2. Experimental data of osmotic pressure for sucrose solutions, compared with Granik and Ferrari, van’t Hoff, and Morse theories and literature data. The regression curve was obtained by applying the maximum likelihood estimation method (correlation coefficient R ) 1), assuming a normal distribution of the measured osmotic pressures for each molality value. A linear model for the standard deviation was adopted; σ decreases from 4% to 2.45% of the estimated mean pressure values, moving from low to high molality values (zero pressure excluded). The confidence interval (1 - R ) 95%) width for the single values varies from (7.8% (at low molality, zero pressure excluded) to (4.8% (molality 5.5) of the corresponding regression curve value. The graph shows the experimental series by Brown and Wolf,30 Garner,31 Morse,32 Money,33 Rau,34 Robinson,35 Moore,36 and Hall.37
TABLE 1: Comparison of the Theoriesa
Materials and Methods
theories
standard deviation S (%)
correlation factor R
Granik et al. Morse van’t Hoff
9.35 33.27 69.9
1 0.995 0.954
a The standard deviation S of the experimental data referred to the theoretical predictions as a percentage of the mean experimental data. The correlation factor R is also shown.
not sufficient to give good estimates of osmotic pressure has been proposed in the literature (e.g., Robinson and Stokes26). Cochrane and Cochrane27 recently provided evidence supporting that the osmotic potential of organic solutions is largely a function of the size of their solute particles. They concluded that solutes influence osmotic potential by altering the molecular spacing of the free water molecules in solution, and therefore different solutes differently influence the osmotic potential. When eq 15 is further simplified, for the case in which a semipermeable membrane separates a pure solvent A from an infinitely diluted binary solution B, the reduced relation coincides precisely with van’t Hoff’s (1) and Morse’s (2) laws. The most detailed theoretical derivation can be found in Granik et al.7 The objective of this work is to evaluate the prediction ability of Granik et al.7 theory, by experimental measurement of membrane osmometry and by meta-analysis of literature data.
Osmotic pressure measurements have been performed for sucrose (Sigma-Aldrich) aqueous solutions in a wide range of concentrations. The high water solubility of sucrose allows measurements to be performed with solutions in a molality range 0-5.5. Bidistilled water was used as the solvent. Polyamide composite reverse osmosis membranes (Sterlitech Corp., Kent, WA) were used to separate the solution from the pure solvent, filtering the solute molecules. These membranes have an average NaCl rejection of 99.5% (nominal MWCO < 150), and operate in a 4-11 pH range and at a maximum temperature of 50 °C. The membrane disks were used at the operating condition range prescribed by the producer. We experimentally verified that sucrose was totally filtered by the membrane. A membrane osmometer able to measure sufficiently high osmotic pressure was not commercially available. That led us to the design and realization of a specific apparatus that was able to support the membranes and prevent them from bursting at pressures up to 20 MPa. It is composed of two cylindrical 303L stainless steel shells in which the solvent and solution chambers are housed. The semipermeable membrane was sealed between the two shells and was held by a sintered stainless steel porous disk to avoid membrane damage or deformation. Further details regarding the device are discussed in Supporting Information.
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J. Phys. Chem. B, Vol. 111, No. 40, 2007 11773
Figure 3. Osmotic pressure data of glucose, maltose, lactose, glycerol, and ethylene glycol solutions compared to the theoretical prediction of Granik and Ferrari Theory, Morse theory, and van’t Hoff theory. The osmotic pressure series were obtained by freezing point depression found in literature.30
TABLE 2: Comparison of the Theoriesa S% glucose lactose maltose glycerol ethylene glycol
van’t Hoff
Morse
Granik et al.
k
R
12.15 6.44 7.49 14.10 23.86
3.57 3.36 2.69 4.05 3.01
2.23 1.16 1.79 0.59 1.40
0.962 1.009 0.946 0.985 1.036
0.9999 0.9997 0.9999 0.9999 0.9988
a The standard deviation S of the experimental data referred to the theoretical predictions as a percentage of the mean experimental data. The fitting coefficients k and their coupled correlation factors R are also shown.
The device components and tools were accurately cleaned with detergent and denatured alcohol and rinsed with distilled water during the set up of each test. The solutions were prepared using an electronic scale (Kern & Sohn GmbH) with an accuracy of (0.01 g. The solution chamber was filled and hermetically sealed to maintain the internal volume at a constant value. The solvent chamber was at atmospheric pressure throughout the entire process and communicated with the outside through a thin graduated PE tube that allowed the volume of solvent that passed through the membrane during each measurement to be measured. Osmotic pressure was continuously monitored with a 25 MPa full scale piezoelectric pressure transducer (AEP transducer). The temperature was constantly controlled; each test was carried out at a temperature of 20 ( 0.1 °C. The final molal concentration was derived by reading the amount of solvent passed through the membrane during each test. An error analysis
(discussed in Supporting Information) was developed to evaluate the maximum measurement error. It decreased from 6% to 1.5% of the measured osmotic pressure data in the molality range 0-1.5 and was accounted for a value of 1.5% in the molality range 1.5-5.5. To evaluate the repeatability of the experimental measures, a statistical analysis was carried out by applying the maximum likelihood estimation method.28 Due to the absence of systematic errors, a normal distribution of the measured osmotic pressures is assumed for each molality value. By maximizing the likelihood function, defined as the product of all probability densities associated with each experimental data, we found the three coefficients of a second-order polynomial that approximates the mean values of the previously hypothesized normal distributions and the standard deviation. A constant and a linear function of molality were initially hypothesized to model the standard deviation σ. The linear model of the standard deviation was identified as the one that best describes the experimental data distribution through the LR-test,29 with a confidence level of 95%. Results and Discussion Sucrose. To obtain a good statistical representation, 43 tests were carried out for the sucrose solution in a 0-5.5 molality range. Figure 2 shows the measured experimental data compared to the theoretical predictions and to other experimental data found in literature.30-37 In the molality range 0-0.7, the three theories predict approximately the same values, with a lower difference than 14%. For higher molality values, the theoretical
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curves diverge, and the experimental data trend is close to the new theory curve. An analysis of the results shows that the deviation of the individual experimental values from the corresponding new theory values is no higher than 17% for any experiment. The experimental data found in the literature shows the same trend as our results within low molality range up to 1.4. The standard deviation S of our experimental data referring to the theoretical predictions is reported in Table 1 as a percentage of the mean pressure data. The correlation factor R is also shown. It can be observed that although the data are positioned very close to the curve corresponding to the new theory, almost all of them are lower than the theoretical predictions. The difference between the experimental data and the theoretical predictions increases with the molality. By multiplying eq 16 by a coefficient k equal to 0.927, the resulting curve overlays the measured data, with a standard deviation of 0.244 MPa corresponding to 2.5%, and a correlation factor R ) 1. The k value was calculated by minimizing the standard deviation of the theoretical values from the experimental data. The small but consistent deviation between experimental data and theoretical predictions is not fully explained at this time. Our current hypotheses include the varied physical-chemical properties of the solutions at high concentration, and atypical mass transport properties owing to interactions between the suspended phase and the pores surfaces38 in the semipermeable membranes. Both hypotheses will be the subject of programs of investigation. Literature Data on Glucose, Lactose, Maltose, Glycerol, and Ethylene Glycol. Experimental data of freezing point depression for aqueous solutions of glucose, maltose, lactose, glycerol, and ethylene glycol were found in the literature.30 The osmotic pressure of these solutions was obtained by taking into account the relationship between solvent activity a1 and freezing point depression39
ln a1 )
[
( )]
-∆Hf(Tf-T) ∆Cf (Tf - T) Tf + - ln RTfT R T T
(17)
where T (K) is the freezing point of solution, Tf (K) is the freezing point of water, ∆Hf ()6008 J/mol) is the latent heat of water, ∆Cf ()38.7 J/(mol·K)) is the heat specific difference between water and ice, and R is the gas constant. The osmotic pressure was calculated by the equation
π)
-RT ln a1 Vw
(18)
where Vw ()18.018 cm3/mol) is the molar volume of water. Figure 3 shows the evaluated osmotic pressure data compared to the theoretical predictions. Glucose and maltose data are very close to the Granik et al.7 curve, but almost all of them are lower than the theoretical prediction. Lactose and glycerol data almost overlay the new theoretical curve. Ethylene glycol data show the same trend as the new theory. The standard deviation S of the osmotic pressure data referring to the theoretical predictions is reported in Table 2 as a percentage of the mean pressure data. By multiplying eq 16 by the coefficients k shown in Table 2, the resulting curves best fit the osmotic pressure data, with correlation factors plotted in the same Table. Conclusions Experimental tests of membrane osmometry were performed for aqueous solutions of sucrose. They were compared, together
with literature data of glucose, maltose, lactose, glycerol, and ethylene glycol to existing fully predictive theories of osmotic pressure. We found that the theory represented by eq 16 shows excellent agreement with experimental observation, in a broad range of concentrations, even in cases where dramatic differences with the classical theories are observed. Acknowledgment. We express our heartfelt gratitude to Prof. Robert Curl for fruitful discussions and to Prof. V. T. Granik, who inspired this work and led the development of its theoretical foundations. We are grateful to Prof. M. P. Calderale and Prof. F. M. Montevecchi for discussions and suggestions. We are indebted to Prof. M. Rossetto for detailed suggestions about statistical analysis. This project has been supported with federal funds from NASA under the contract of SA23-06-017 and Department of Defense under the contract of W81XWH04-2-0035, as well as fund from State of Texas, Emerging Technology Fund. Supporting Information Available: Detailed description of the membrane osmometer and results of the experimental test error analysis. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Hammel, H. T.; Schlegel, W. M. Osmosis and solute-solvent drag - Fluid transport and fluid exchange in animals and plants. Cell Biochem. Biophys. 2005, 42, 277-345. (2) Sakallioglu, E. E.; Ayas, B.; Sakallioglu, U.; Yavuz, U.; Acikgoz, G.; Firatli, E. Osmotic Pressure and Vasculature of Gingiva in Experimental Diabetes Mellitus. J. Periodontol. 2007, 78, 757-763. (3) Verma, R. K.; Arora, S.; Garg, S. Osmotic pumps in drug delivery. Crit. ReV. Ther. Drug. 2004, 21, 477-520. (4) Su, Y. C.; Lin, L. W. A water-powered micro drug delivery system. J. Micro-Electromech. Syst. 2004, 13, 75-82. (5) van’t Hoff, J. H.; Die Rolle der osmotischen Druckes in der Analogie zwischen Lo¨sungen und Gasen. Z. Phys. Chem. 1887, 1, 481508. (6) Morse, H. N.; Frazer, J. C. W.; Rogers, F. M. The osmotic pressure of glucose solutions in the vicinity of the freezing point of water. J. Am. Chem. 1907, 38, 175. (7) Granik, V. T.; Smith, B. R.; Lee, S. C.; Ferrari, M. Osmotic Pressures for Binary Solutions of Non-electrolytes. Biomedical MicrodeVices 2002, 4:4, 309-321. (8) Guttman, H. J.; Anderson, C. F.; Record, M. T. Analyses of thermodynamic data for concentrated hemoglobin-solutions using scaled particle theory - implications for a simple 2-state model of water in thermodynamic analyses of crowding in-vitro and in-vivo. Biophys. J. 1995, 68, 835-846. (9) Cohen, J. A.; Highsmith, S. An improved fit to Website osmotic pressure data. Biophys. J.1997, 73, 1689-1694. (10) Kalyuzhnyij, Y. V.; Rescic, J.; Vlachy, V. Analysis of osmotic pressure data for aqueous protein solutions via a one-component model. Acta Chim. SloV. 1998, 45, 194-208. (11) Yokozeki, A. Osmotic pressures studied using a simple equationof-state and its applications. Appl. Energ. 2006, 83, 15-41. (12) Zhao, Y.; Taylor, J. S. Incorporation of osmotic pressure in an integrated incremental model for predicting RO or NF permeate concentration. Desalination 2005, 174, 145-159. (13) Ahlqvist, J. Equation for osmotic pressure of serum protein (fractions). J. Appl. Physiol. 2004, 96, 762-764. (14) Wu, J.; Prausnitz, J. M. Osmotic pressures of aqueous bovine serum albumin solutions at high ionic strength. Fluid Phase Equilib. 1999, 155, 139-154. (15) Su, Y. H.; Riffat, S. B.; A thermodynamic approach to calculating the operating osmotic pressure of pressure-driven membrane separation absorption cycles. Int. J. Therm. Sci. 2004, 43, 1197-1201. (16) Janacek, K.; Sigler, K. Osmosis: Membranes impermeable and permeable for solutes, mechanism of osmosis across porous membranes. Physiol. Res. 2000, 49, 191-195. (17) Kim, K. S.; Davis, I. S.; Macpherson, P. A. et al. Osmosis in small pores: a molecular dynamics study of the mechanism of solvent transport. Proc. R. Soc. London, Ser. A-Math. Phys. Eng. Sci. 2005, 461, 273-296.
Osmotic Pressure beyond Concentration Restrictions (18) Kargol, M.; Kargol, A. Mechanistic Formalism for Membrane Transport Generated by Osmotic and Mechanical Pressure. Gen. Physiol. Biophys. 2003, 22, 51-68. (19) Ben-Sasson, S. A.; Grover, N. B. Osmosis: A macroscopic phenomenon, a microscopic view. AdV. Physiol. Educ. 2003, 27, 15-19. (20) Kargol, M.; Suchanek, G.; Przestalski, M. et al. The problem of water exchange by living cells in the light of mechanistic transport equations. Pol. J. EnViron. Stud. 2005, 14, 605-611. (21) Prausnitz, J. M. Molecular thermodynamics for some applications in biotechnology. Pure Appl. Chem. 2003, 75, 859-873. (22) Green, N. M.; Channels, Carriers and Pumps - an Introduction to Membrane Transport - Stein, W. D. Nature 1991, 351, 450-451. (23) Robinson, C. R.; Sligar, S. G. Participation of water in Hin reconbinase - DNA recognition. Protein Sci. 1996, 5, 2119-2124. (24) Kornblatt, J. A.; Kornblatt, M. J. The effects of osmotic and hydrostatic pressures on macromolecular systems. Biochim. Biophys. Acta 2002, 1595, 30-47. (25) Shimizu, S. Estimating hydration changes upon biomolecular reactions from osmotic stress, high pressure, and preferential hydration experiments. P. Natl. Acad. Sci. U.S.A. 2004, 101, 1195-1199. (26) Robinson, R. A.; Stokes, R. H. Electrolytic Solutions; Butterworths: London, 1970. (27) Cochrane, T. T.; Cochrane, T. A. Osmotic Properties of Organic and Inorganic Solutes and their Influence on Flow at Different Stages of the Soil-Plant Solution Continuum. Soil Sci. 2007, 172, 386-395. (28) Gil’t, I. Y.; Vinichenko, S. M.; Sysoev, Y. S. General problems of metrology and measurement technique - The construction of the optimum group strategy for different distribution laws of the errors of expert estimates. Meas. Technol. 2005, 48, 1021-1029.
J. Phys. Chem. B, Vol. 111, No. 40, 2007 11775 (29) Goglio, L.; Rossetto, M. Comparison of fatigue data using the maximum likelihood method. Eng. Fract. Mech. 2004, 71, 725-736. (30) Brown, M. G.; Wolf, A. V. Aqueous Solutions and Body Fluids; Hoeber: New York, 1966. (31) Garner, W. E. Osmotic pressure. International Critical Tables of Numerical Data, IV; McGraw-Hill: New York, 1928; pp 429-432. (32) Morse, H. N.; Holland, W. W.; Myers, C. N.; Cash, G.; Zinn, J. B. Am. Chem. 1912, 48. (33) Money, N. P.; Osmotic pressure of aqueous polyethylene glycols - Relationship between molecular weight and vapor pressure deficit. Plant Physiol. 1989, 91, 766-769. (34) Rau, D. National Institutes of Health, NIDDK and DCRT, Laboratory of Structural Biology, Bethesda, MD 20892-0580 http://www.brocku.ca/researchers/peter_rand/osmotic/osfile.html. (35) Robinson, R. A.; Stokes R. H. Tables of osmotic and activity coefficients of electrolytes in aqueous solution at 25°C. Faraday Soc. Trans. 1949, 45, 612-624. (36) Moore, W. J. Physical chemistry, 4th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1972; pp 250-253. (37) Hall, R. E.; Sherrill, M. S. Freezing-point lowerings of aqueous solutions. International Critical Tables. McGraw-Hill: New York, 1928; Vol. VI, pp 254-264. (38) Pricl, S. et al. Multiscale modeling of protein transport in silicon membrane nanochannels. Part 1. Derivation of molecular parameters from computer simulations. Biomed. MicrodeV. 2006, 8, 277-290. (39) Gu, X.; Suzuki, T.; Miyawaki, O. Limiting Partition Coefficient in Progressive Freeze-concentration. J. Food Sci. 2005, 70, E546-E551.
