2341
Ind. Eng. Chem. Res. 1994,33,2341-2344
Altering Diffusivities in Dilute Polymeric and Biological Solutions John L. Gainer Department of Chemical Engineering, University of Virginia, Thornton Hall, Charlottesville, Virginia 22903-2442
Diffusion can often be the limiting factor in determining the rate of a liquid-phase process. On the basis of absolute rate theory, Metzner and co-workers developed theoretical expressions for the prediction of diffusivities in viscous liquids and polymer solutions. Those equations are summarized here, and this approach is modified for the case of dilute polymeric solutions, including biological ones, in order to gain insight into the mechanisms involved in altering diffusion rates. Increasing the diffusivity may be especially beneficial in biological systems, where small changes might have a significant effect. For example, it is suggested that increasing the diffusivity of oxygen could alter (1) the respiration rate of microorganisms in culture and (2) the rate of oxygen consumption in animals, and data for these are presented.
Background Mass transfer by diffusion occurs in many processes, and, frequently, is the factor which is rate-limiting. This appears to be especially true for biological systems. According to Sherwood, Pigford, and Wilke (1975, p 1) ”mass-transfer phenomena are to be found everywhere in nature...”. Another classictext, Lehninger’s Biochemistry (1975, p 179), states that diffusion may be responsible for the very way that living systems have developed. “Diffusion is a fundamental process in all cellular transport activities. The rate of diffusion and the diffusion-path length of various metabolites and enzymes are believed to set physical limits on the size and volume of the metabolizing mass of living cells and their organelles.” Thus, in order to analyze or alter biological processes it would seem that diffusion effects should be considered first; however, this is seldom the case. The engineer also frequently encounters systems in which the rate of a chemical reaction is limited by diffusion. As is well-known from Fick‘s law, diffusive flux is proportional to the product of the diffusion coefficient, or the diffusivity, and the local concentration gradient (Sherwood et al., 1975). Usually, considerable effort is expended to increase the flux through increasing the gradient, either by reducing the path length or by increasing the concentration driving force. Increasing the diffusivity should have a similar effect, but this approach has been overlooked. Indeed, it is often assumed that the diffusion coefficient is a constant and that only an orderof-magnitude estimate of its value is needed for designing a process. However, it should also be possible to change the rate of a diffusion-controlled process by increasing or decreasing the diffusivity, and this may be especially important in living systems. In order to develop methods for altering the diffusion coefficient, though, it is imperative to have an adequate model for the diffusion process as well as a method for predicting the diffusivity. Several models have been developed previously which are intended to do this, and for diffusion through liquids it is possible to classify them as arising from (1) hydrodynamical considerations, (2) extensions of kinetic theory of gases to liquids, and (3) absolute rate theory. The hydrodynamical approaches to the prediction of the diffusivity are based on the Nernst-Einstein equation (Daniels and Alberty, 1956) which relates the movement of a particle through a liquid medium to a “resisting force” which retards such motion. When this resisting force is 0888-5885/94/2633-2341$04.50/0
approximated using the Stokes drag on a particle, the much-used Stokes-Einstein model results:
where D m is the diffusivity of solute A through liquid B, k is the Boltzmann constant, Tis the absolute temperature, PB is the viscosity of liquid B, and RAis the radius of solute A. This equation has been used frequently, but gives accurate predictions of the diffusivity only if the solute is a large spherical molecule diffusing in a medium which appears as a continuum to the diffusing species. Thus, although this equation is the one most often cited by the life scientists, it does not describe the case where the diffusing species is of the same size, or smaller, than the molecules of the fluid through which it is diffusing. It is this situation, of course, which is frequently encountered in living systems, such as in the diffusion of oxygen though a liquid like blood or the cytoplasm of a cell. In order to predict diffusivities of molecules dissolved in liquids, an empirical modification of eq 1was made by Wilke and Chang (1955):
D,
(7.4 x 10-8)(X~M~)0’5T =
(2)
PBV:O
where D m is in units of cm2/s,M B is the molecular weight of liquid B, and VA is the molar volume of the solute A. XB is an “association number” which has a value of 1 for nonassociated solvents, but has other values for solvents such as water and alcohols, where significant hydrogen bonding occurs. This equation predicts diffusivities well for some liquids, but of course, it is necessary to know the value of the association number, a priori, which somewhat limits its usefulness. In addition, eq 2 does not accurately predict the diffusivity in viscous liquids and polymer solutions, including biological fluids. Other equations of the same general type as the Wilke-Chang equation have also been proposed (Scheibel, 1954;Sitamaran et al.,1963; Reddy and Doraiswamy, 1967), in which the diffusivity varies inversely with the viscosity raised to a power which ranges from 1/3 to 2/3. The treatment of liquids as condensed gases is based on the kinetic theory of gases and is limited to very simple liquids; thus, it will not be described here. Absolute rate theory, however, has been applied to all types of systems. It is based on the original Eyring rate theory (Glasstone et al., 1941), which explains both viscosity and diffusion 0 1994 American Chemical Society
2342 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
on the basis of a lattice model for the liquid which contains vacancies,or holes. A certain “activation energy”is needed for either process. In the case of diffusion, transport can be envisioned as the solute jumping from hole to hole in the liquid lattice. One of the first major modifications to this approach was made by Gainer and Metzner (1965), when they presented the following equation for the prediction of the diffusivity:
(V, of the solution, S, and of the solvent, B, as well as the molecular weight ratio (MdMs)usually cancel each other, resulting in a value close to unity. Therefore, the diffusivity of a solute A in a solution, S, divided by the diffusivity of the same solute in the solvent B is essentially determined by the value of AE, which is the difference in the activation energies for diffusion (AE= ED^ - ED,). It was reported that this activation energy difference is related (1) to the solution viscosity (Li and Gainer, 1968) or (2) to the solution viscosity plus a factor which accounts for the “stiffness”of the polymer molecules (Navari et al., 1971). Using the latter approach, the activation energy difference can be approximated by
where AE = AE,,
5 represents the number of nearest neighbor molecules to the diffusing molecule, N is Avogadro’s number, r f i and ?‘BB are the distances between two similar molecules (estimated from the cube root of the molar volume divided by Avogadro’s number), rm is the distance between two dissimilar molecules (estimated using the arithmetic average of r u and QB), E r is~ the activation energy for viscosity, and ED- is the activation energy for diffusion. E, is the sum of two components, that due to hydrogen bonding, E,,H, and that due to dispersion forces, E,,D. The ratio of the contribution due to hydrogen bonding relative to that due to dispersion forces is estimated to be the same as the ratio of the heat of vaporization due to hydrogen bonding to the total heat of vaporization as given by others (Bondi and Simkin, 1957). E, can be calculated using a modified form (Gainer and Metzner, 1965) of the original equation given by Glasstone, Laidler, and Eyring (1941) as
where p is the viscosity, V is the molar volume, AEvapis the energy of vaporization, M is the molecular weight, and Tis the temperature. It was shown (Gainer and Metzner, 1965) that the use of this equation results in accurate predictions for the diffusivities of solutes in many liquids. It is especially useful for viscous systems where it results in predictions of the diffusivity that are within *20% of the experimental value (where previous equations were in error by an order of magnitude). As is obvious, these equations are more complicated than previous ones, and, even though they contain no adjustable parameters, are more difficult to use. However,since this approach proved to be relatively accurate for diffusion in viscous liquids, it seemed reasonable to extend its use to diffusion in polymer solutions. Several modifications to absolute rate theory were proposed at about the same time in order to predict diffusivities in polymer solutions. It was suggested (Li and Gainer, 1968; Navari et al., 1971) that the diffusivity in a polymer solution could be compared to that in the pure solvent, resulting in the following equation (Navari et al., 1971):
(6)
where AEm, is related to intermolecular distances (asare the r u , QB and rm terms in eq 31, c is the polymer concentration, and k is related to the intrinsic viscosity of the solution. For very dilute polymer solutions, k might be assumed to be zero; thus, for those cases, the diffusivity can be predicted using some measure of the “structure” of the solution. One of the first attempts to quantify this was by Osmers and Metzner (1972). They noted that the addition of polymer molecules to solvents should alter the intermolecular spacing in the liquid, affecting the activation energy difference. They further suggested that evaluating the change in the specific volume after mixing the polymer and solvent together (AV-) should provide an indication of this change in liquid structure, and observed that both the diffusivity and AVm, in polymer solutions usually decrease as the polymer concentration increases. Although they did not show a correlation for this, the data they presented will form the basis for the results shown in the next section.
Theory and Results Equations 5 and 6 indicate that, for dilute polymer solutions, the ratio of the diffusivity of a solute though a solution relative to the diffusivity of that same solute through the solvent alone is related only to a measure of the intermolecular distances. Following the reasoning of Osmers and Metzner, AE should be related to AVm,. For example, it could be assumed that AE is directly proportional to AVmix, or
AE = k’AV,, which results in eq 5 being written as
DAslD, = exp(aAVm,)
(7)
where a! is equal to k‘IRT. It is possible to write the exponential term in eq 7 as a series expansion, and for small values of the excess volume of mixing, that series can be truncated to give (8)
The excess volume of mixing of a polymer solution, AVmix, can be calculated if the densities of the polymer, pop, the solvent, OB, and the solution, ps, are known, as well as the weight fractions of the polymer and the solvent, wp and WB:
The preexponential terms containing the molar volumes
exp(-klc)
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2343
l.2{
9s DAB
'4 /
9s
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Figure 1. Diffusivity of solutes through polymer solutions versus excess volume of mixing of the solutions.
By use of the data presented in the paper by Osmers and Metzner (1972) for volume changes upon mixing in polymer solutions and the diffusivity of a solute through these solutions, Figure 1 is obtained. In Figure 1, the letter A stands for the solute, B for the solvent, and S for the solution. It is interesting to note that, as has been previously suggested by others (Navari et al., 1971), using the ratio of the diffusivity of the solute through the solution, D M ,relative to the diffusivity of that same species through only the solvent, DAB,results in the canceling out of any effect of the solute. Thus, the data for the diffusion of any solute can be correlated on the same plot. As Osmers and Metzner (1972) noted, the diffusivity decreased in the polymer solutions relative to that in the pure solvents, resulting in a value of DMIDAB less than unity. The volumesof mixing obtained for the same polymer solutions were negative values, and the correlation shown in Figure 1 appears to be linear as eq 8 suggests. Only decreases in the diffusivity were seen as a polymer was added to a solvent. However, Gainer and co-workers (Chisolm and Gainer, 1974; Gainer and Brumgard, 1982; Gainer et al., 1993) also used this approach in order to find a way to increase the diffusivity of oxygen in a biological polymer solution, blood plasma. Although plasma contains about 7 wt ?6 of proteins in water, along with numerous small molecules such as amino acids and glucose,the viscosity is not that much different from water, leading to the assumption that k in eq 6 is small. Thus, if only the intermolecular spacing is important, adding a substance to the plasma which causes an increase in the excess volume of mixing should result in an increased diffusivity of oxygen. It had been previously suggested that the length of a rigid segment was of importance (Navari et al., 1971): a long "stiff" molecule would cause an increase in the volume of mixing. The carotenoid crocetin was chosen as such a "stifr compound, due to its conjugated carbon-carbon bonds. The volumes of mixing were obtained for crocetincontaining plasma solutions, as were the diffusivities of oxygen in those same solutions. Those data are shown in Figure 2, as well as data for polymer solutions shown in Figure 1. I t can be seen that all of the data are reasonably well correlated by a straight line. In addition, Figure 2 shows that the ratio of the diffusivities is unity when the excess volume of mixing is zero, consistent with eq 8. Thus, it appears that the value of the excess volume of mixing allows one to predict how the diffusivity is affected though dilute polymer solutions. This simple relationship does not hold for concentrated polymer solutions where the effect on the viscosity must also be included. Such a correction can be made (Navari et al., 1971), and diffu-
0
(x I@),
.
, . ,
. J
2
3
1
Cm3/g
Figure 2. Diffusivityof solutae throughpolymer solutions,including blood plasma containingcrocetin, versus excess volume of mixing of the solutions. Table 1. Effect of Crocetin on Respiration Rates of Microorganisms Streptomyces griseus, T = 30 O C , Emerson's Broth, 5 % Glucose respiration rata, (mg of Oz/(min.mgof cell sample dry weight)) X 104 control 2.68 crocetin added 3.40" Bacillis subtilis, 37 "C, Nutrient Broth respiration rate, (mg of Oz/(min.mgof cell sample dry weight)) X 1010 control 1.24 crocetin added 1.66b a crocetin caused an increase in respirationrate of 27 % caused an increase in respiration rate of 34%.
.
~~
Croceth
sivities in concentrated polymer solutions can then be predicted using this same approach. It has been more common, however, to use free-volume theories for the prediction of mutual diffusion coefficients in concentrated polymer solutions (Duda et al., 1982). Actually these two approaches are somewhat similar to each other, although the computations can be complicated in either case and may require considerable data. However, naturallyoccurring, polymeric, biological solutions are almost always fairly dilute, so the simpler, modified approach suggested here might be useful in altering the mass transport in such systems. As noted earlier, less than an order-of-magnitude change in the value of the diffusivity is often ignored by engineers. However, some systems, such as biological ones, where diffusion has been said to be of major importance, may be greatly affected by small changes in the diffusivity. For example, the use of the compound crocetin has been found to increase the diffusivity of solutes, such as oxygen, through aqueous-based solutions by 20-30 ?6 (Gainer and Brumgard, 1982; Gainer et al., 1993). If certain natural processes are diffusion-controlled, their rates should increase correspondingly by 20-30% with the use of crocetin. This has been tested for two different cases, (1) the respiration of microorganisms and (2) the oxygen consumption of an animal after blood loss (hemorrhage). Respiration rates of two microorganisms, one a bacterium and one a fungus, are shown in Table 1. These were measured (King and Gainer, 1977) using the standard method of abruptly discontinuing the aeration of a microbial culture, and determining the time dependence of the oxygen concentration in the solution after that. As
2344 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 Table 2. Whole-Body Oxygen Consumption Rates in Hemorrhaged Rats oxygen consumption, weight, g mL/(min.kg) Effect of Hemorrhage 21.1 i 0.8 no hemorrhage 8 298f 14 6 299 f 19 14.2 0.5 40% hemorrhage Effect of Crocetin After Hemorrhage 6 299 f 19 14.2 f 0.5 40% hemorrhage 6 292 f 15 18.3 f 0.5" 40% hemorrhage plus crocetin mOUR
N
*
Crocetin results in a 29% increase in oxygen consumption.
can be seen, microbial respiration rates were increased with crocetin by 25-35 % ,which is about the same amount by which the diffusivity is increased. Table 2 shows the effect of crocetin on oxygen consumption rates in hemorrhaged animals (Gainer et al., 1993). Whole-body oxygen consumption decreases markedly with blood loss (which results in the onset of shock), and this is shown in the top of Table 2. However, the use of crocetin resulted in a 30% increase in the oxygen consumption rates, and again, the increase is the same amount that the diffusivity is increased (see the lower part of Table 2). Extensive tests (Gainer et al., 1993) were done to rule out any other effect that the crocetin might have had on oxygen transport from the blood or utilization by the tissues in the body. There is not general agreement that either the respiration of microorganisms or the oxygen consumed by the body tissues are diffusion-controlled. In both cases, however,the characteristic Reynolds number is very small, leading to the common assumption that the mass transfer coefficient is proportional to the diffusivity of the solute. It is striking that both rates are increased by a similar amount by the addition of crocetin, and this increase corresponds, approximately, to the percentage change in the diffusivity. These results are presented here to suggest that small changes in the diffusivity may have important consequences in certain cases, especially in biological systems. For example, the 30% increase in oxygen consumption after hemorrhage resulted in a much increased chance of survival after shock (Gainer et al., 1993), as had beensuggested by others (Crowelland Smith, 1964; Wilson et al., 1972). In summary, the previous results of Metzner and coworkers have been extended to new systems. Diffusivity changes may be estimated by simply measuring the density of the solution after a compound is added and calculating the excess volume of mixing, as suggested by Osmers and Metzner (1972). Again, it should be emphasized that this simplification does not hold for concentrated polymer
solutions. For those systems, the effect of viscosity changes on the diffusivity must be accounted for as indicated by previous studies (Li and Gainer, 1968; Navari et al., 1971; Osmers and Metzner, 1972). Literature Cited Bondi, A.; Simkin, D. J. Heats of Vaporization of Hydrogen-bonded Substances. AZChE J. 1957,3,473-479. Chisolm, G. M.; Gainer, J. L. Oxygen Diffusion and Atherosclerosis. Atherosclerosis 1974,19, 135-138. Crowell, J. W.; Smith, E. E. Oxygen Deficit and Irreversible Hemorrhagic Shock. Am. J. Physiol. 1964,206,313-316. Daniels, F.; Alberty, R. A. Physical Chemistry; John Wiley & Sons, Inc.: New York, 1956. Duda, J. L.; Vrentas, J. S.;Ju, S. T.; Liu, H. T. Prediction of Diffusion Coefficientafor Polymer-Solvent Systems.AZChE J.1982,28,279285. Gainer, J. L.; Metzner, A. B. Diffusion in Liquids-Theoretical Analysis and Experimental Verification.AZChE-Z.Chem.E.Symp. Ser. 1965,6, 74-82. Gainer, J. L.; Brumgard, F. B. Using Excess Volume of Mixing to Correlate Diffusivities in Liquids. Chem. Eng. Commun. 1982,15, 323-329. Gainer, J. L.; Rudolph, D. B.; Caraway, D. L. The Effect of Crocetin on Hemorrhagic Shock in Rats. Circ. Shock 1993,41,1-7. Glasstone, S.; Laidler, K. J; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. King, M. L.; Gainer, J. L. Oxygen Diffusion and Fermentations. AZChE Symp. Ser.: Water-1976 1977,167, 1-5. Lehninger, A. L. Biochemistry, 2nd ed.; Worth Publishing, Inc.: New York, 1975. Li, S. U.; Gainer, J. L. Diffusion in Polymer Solutions. Ind. Eng. Chem. Fundam. 1968, 7,433-440. Navari, R. M.;Hall, K. R.; Gainer, J. L. Predictive Theory for Diffusion in Polymer and Protein Solutions. AZChE J.1971,17,1028-1036. Osmers,H. R.; Metzner, A. B. Diffusion in Dilute Polymeric Solutions. Znd. Eng. Chem. Fundam. 1972,11,161-169. Reddy, K. A.; Doraiswamy,L. K. Estimating Liquid Diffusivity. Znd. Eng. Chem. Fundam. 1967,6, 77-79. Scheibel, E. G. Liquid Diffusivities. Znd. Eng. Chem. 1954,46,20072008. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Sitamaran, R.; Ibrahim, S. H.; Kuloor, N. R. A Generalized Equation for Diffusion in Liquids. J. Chem. Eng. Data 1963,8, 198-201. Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficienta in Dilute Solutions. AIChE J. 1955, 1, 264-270. Wilson, R. F.; Christensen, C.; Leblanc, L. P. Oxygen Consumption in Critically I11 Patients. Ann. Surg. 1972, 176, 801-804. Received for review November 15, 1993 Revised manuscript received February 22, 1994 Accepted March 10, 1994' ~~~
* Abstract published in Advance ACS Abstracts, August 15, 1994.
