1068
J. Phys. Chem. 1984,88, 1068-1076
charge-transfer character. Even if that equation does constitute a good approximation to AGO for the quenching process, data for oxidation and reduction potentials are not available in toluene. As a rule, reduction potentials become more negative and oxidation potentials more positive with a decrease in solvent polarity;'* i.e., the transfer of charge becomes more difficult. On the other hand, the Coulomb stabilization, eZ/crDA in eq 1, will also increase in nonpolar solvents. Though the two effects cancel each other to some extent, still one cannot predict the free energy change for the quenching process in toluene on the basis of its value in a polar solvent. Nevertheless, for lack of the correct AGO values, log k, in toluene in Figure 1 was plotted against AGO as calculated for acetonitrile. In view of this crude representation the similarity in the quenchihg rate constants in the two solvents is quite remarkable. For quenchers with very negative AGO values for electron transfer, i.e. good electron donors, in acetonitrile, it stands to reason that also in toluene the free energy change for the process will still remain negative. The data points ought perhaps be shifted along the AGO axis but would still remain in the AGO < 0 range. The measured reaction rates in toluene in these cases show that the process here too is encounter limited. A mechanism invoking charge transfer to account for the fluorescence quenching of ZnOEP by electron donors and acceptors in both polar and nonpolar solvents agrees well with the obserations of Ballard and Mauzerall,' who found by conductometric measurements that the rate of the ZnOEP triplet-ZnOEP ground-state reaction yielding positive and negative ions is very similar to those in CH3CN and (12) Peover, M. "Electroanalytical Chemistry"; Bard, A. J., Ed.; Marcel Dekker: New York Vol. 2. ChaDter IIC. (13)."Techniques of Electroorganic Synthesis" Part 11; Weinberg, N. L., Ed.; W h y : New York, 1975; Vol. 5, appendix. (14) Meites, L.; Zusman, P. "Electrochemical Data"; Wiley: New York, London, 1973; Part I, Vol. A. (15) Suatori, J. C.; Snyder, R. E. Anal. Chem. 1961, 33, 1894-1897.
in toluene. Of course, the free ion yields differ vastly in the two solvents. It is even more astonishing that when the electrochemically determined AGO values (eq 1) in acetonitrile become positive and when they might have a quite different value in toluene, the quenching constants for ZnOEP in the two solvents still resemble each other. The similar position of the break in the log k, vs. AGO curve leads us to assume that the contribution to the true AGO values due to electron transfer in toluene and in acetonitrile are not very different. The log k, values in the two solvents in the range where log k, < 1O'O indicate that the quenching mechanisms, which might differ for different quenching substances, are apparently similar for a given quencher in the two solvents. This is brought out by the data for substances 14, 15, and 16 in Figure 1. In conclusion, we feel it is very likely that as long as the electrochemicallydetermined AGO value is negative, the quenching of ZnOEP fluorescence in both polar and apolar media takes place via electron transfer. For quenching substances with redox potentials such that AGO > 0 for the transfer of an electron, other quenching routes might become more efficient. However, in such cases the rate-determining step in the reaction between ZnOEP and the quencher seems to be the same in both the polar acetonitrile and the apolar toluene.
Acknowledgment. This research has been sponsored in part by the U S . Army through its Research and Standardization Group (Europe). Registry No. ZnOEP,17632-18-7; CHjCN, 75-05-8;toluene, 10888-3;p-benzoquinone, 106-51-4; 1,4-naphthoquinone, 130-15-4; duroquinone, 527-17-3;9,10-anthraquinone,84-65-1; benzaldehyde, 100-52-7; benzoic acid, 65-85-0;acrylamide, 79-06-1;naphthalene, 91-20-3;phenylenediamine, 25265-76-3; o-aminophenol, 95-55-6;a-naphthylamine, 134-32-7; 3,4-dimethylphenol, 95-65-8; N,N-diethylaniline, 91-66-7; a-naphthol, 90-15-3; 1,4-hydroquinone, 123-31-9;phenol, 108-95-2.
Kinetlcs of Multicomponent Transport by Structured Flow In Polymer Solutions. 5. Ternary Diffusion in the System Poly(vinylpyrro1idone)-Dextran-H,O W. D. Cornper,* G . J. Checkley, and B. N. Preston Biochemistry Department, Monash University, Clayton, 3168, Victoria, Australia (Received: April 26, 1983; In Final Form: July 18, 1983)
The etiological events associated with the formation of structured flows in dextran-poly(vinylpyrro1idone) systems have been investigated. The numerical values of all four diffusion coefficients in the model ternary system have been calculated from thermodynamic (osmotic) and binary or tracer transport experiments. Utilizing this information and the known mathematical solutions to the system of flow equations has led to the calculation of the spatial density distribution as a function of time. These calculations demonstrate a direct correlation between the polymer concentrations required for the formation of a density inversion at the boundary and the onset of structured flows. Previous theoretical treatments of the stability of these solutions have also been analyzed.
Introduction It has been previously shown that ternary systems consisting of a uniform concentration of dextran (regarded as a pseudosolvent) with an imposed concentration gradient of poly(viny1pyrrolidone) (PVP) exhibit two striking features which are as follows: (i) the transport of either radioactively labeled dextran or PVP in the system is extremely rapid as compared to its difand (ii) this rapid fusional transport in the binary (1) B. N. Preston, T. C. Laurent, W. D. Comper, and G . J. Checkley, Nature (London),287,499 (1980). (2) T. C. Laurent, B. N. Preston, W. D. Comper, G. J. Checkley, K. Edsman, and L.-0. Sundelof, J . Phys. Chem., 87,648 (1983).
0022-3654/84/2088-lO68$01.50/0
polymer transport is accompanied by the formation of visible, coherent structures (structured flows) in the solution as seen by labeling either of the polymers with blue dye.'s4 The onset of rapid transport and structured flow formation has been equated with the critical concentration of the dextran-solvent at which transient network formation occur^.^ (3) B. N. Preston, W. D. Comper, T. C. Laurent, G. J. Checkley, and R. G. Kitchen, J . Phys. Chem., 87,655 (1983). (4) W. D. Comper, B. N. Preston, T. C. Laurent, G . J. Checkley, and W. H.Murphy, J . Phys. Chem., 87, 667 (1983). ( 5 ) B. N. Preston, W. D. Comper, G . J. Checkley, and R. G . Kitchen, J . Phys. Chem., 87,662 (1983).
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 6, 1984 1069
Multicomponent Transport The etiological events associated with the formation of structured flows are complex and little is known. We have previously suggested4 that the initial behavior of components, involving normal diffusion, may lead to boundary instabilities in the form of a density inversion which is then acted on by gravity to ultimately produce structured flows. These instabilities arise though a complex range of dynamic and equilibrium interactions between components as embodied in large coupled diffusion effects. The study presented in this paper is designed to test this model. It was previously shown that the onset of rapid transport of PVP, together with the appearance of structured flows, was dependent on the dextran molecular weight and con~entration.~ For dextran with aW 10000 the onset of rapid transport was apparent at dextran concentrations between 45 and 55 kg m-3 whereas for a higher molecular weight dextran with 500000 rapid transport occurs at concentrations > 5 kg m-3 (ref 5). We have now studied this latter system in detail as rapid transport occurs in relatively dilute solution in this case. The molecular interactions between solutes may then be satisfactorily described by bimolecular interaction’terms. We have measured all the binary interactions in the dextran:PVP:HzO system and have estimated all ternary diffusion coefficients over a wide range of dextran concentration. These data may then be used to predict macroscopic density profiles through the boundary at any time. This has led to the identification of a density inversion at the boundary prior to structured flow formation.
-
coefficients such that L l z = LZl in accordance with the Onsager reciprocal relationship. The phenomenological coefficients may be expressed in terms of ternary diffusion coefficients and chemical potentials through the following relations: ’ -Dll = aLll bLI2 (7)
-D12
+ = ~ L l l+ dL12
(8)
-D21 = aLzl + bLzz
(9)
-Dzz = C L ~+, dLz2
(10)
where the expressions for a, b, c, and d are
-
aW
Theory Diffusion in Ternary Systems. The isothermal diffusion in a ternary system can be completely described by two flow equations along the x coordinate only:
where Z
e = - c1
0 - 3 ~ ~ ~ ~
i=l
In these equations the flows (Ji),are referred to the volume-fixed reference frame, the concentrations mi and ci are in molal and molar, respectively, Mi is the molecular weight of i, and p is the density (Mg m-3). The Dii are the principal or main diffusion coefficients. The Dij are the interaction or cross diffusion coefficients. We assume that for the small concentration differences in the experiments described here, the volume-fixed frame is identical with the cell-fixed (or apparatus-fixed) reference frame in which experimental measurements are made. The subscripts 1, 2, and 3 refer to dextran, PVP, and solvent (water), respectively. Solutions for (Dij)!as functions of phenomenological coefficients and thermodynamic activities have been given by Miller.6 With the condition of zero volume flow where is the molar volume of i, exact expressions for the fluxes of components 1 and 2 may be written as quantities independent of the forces involved, so that the linear laws to which the Onsager reciprocal relationship applies may be written as
c=
[
(1
+$
L
1
2
+ mz Vl
(13)
where pij = api/am,. Further modification of these equations comes from reexpressing the phenomenological coefficients as functions of the binary frictional coefficients. We shall use Spiegler’s’ form of the frictional coefficient (see also ref 8), so that the chemical potential gradient, aptlax, can be written as linear combinations of the relative velocities (vi) and the binary frictional coefficient Gj)per mole of i, so that
Note that since the frictional coefficient is a function of the relative velocity of i and j it becomes independent of the frame of reference. The phenomenological coefficients expressed as functions of the binary frictional coefficients are given in Appendix 1. Experimental Protocol for Evaluation of Ternary Diffusion Coefficients. Major interest in the analysis of ternary diffusion for ths system of PVP-dextran-water resides in understanding the initial events that ultimately yield structured flows. It is assumed that immediately after boundary formation a ternary diffusional regime will proceed until a point of instability is reached upon which the system transforms into a structured flow regime. This transition occurs extremely rapidly after boundary formation. Therefore, classical analysis of ternary diffusion coefficients9using these systems cannot be made. Rather, we have calculated the ternary diffusion coefficients expressed in eq 7-10 through the experimental evaluation of all p y terms from osmotic pressure and phase separation experiments and all& (and therefore LV)from diffusion experiments. These methods will be discussed in detail in the following section. Thermodynamic Parameters. Convenient algebraic expressions for the chemical potential of components 1 and 2 as a function of composition of a mixture of 1 and 2 have been given by Ogston:1° pl - plo =
RT(1n m l
+ (a2),m1+ (a3)1ml2+ a+m2+ ...) (16)
where
[ :ti]
y,=-c6 i j + 2
j=1
-
where (a& and (a3)iare the coefficients expressing thermodynamic
+j
ax
where 6ij is the Kronecker delta and Lij are the phenomenological ( 6 ) I).G. Miller, J . Phys. Chem., 63, 570 (1959).
(7) K . S.Spiegler, Trans. Faraday SOC.,54, 1409 (1958). (8) 0. Kedem and A. Katchalsky, J . Gen. Physiol., 45, 143 (1961); for earlier treatment see 0. Lamm, J . Phys. Chem., 61, 948 (1957). (9) See, for example, E. L. Cussler, ‘Multicomponent Diffusion”, Elsevier, Amsterdam, 1976. (10) A. G. Ogston, Arch. Biochem. Biophys., Suppl. 1, 39 (1962).
1070 The Journal of Physical Chemistry, Vol. 88, No. 6,1984
nonideality and a+ is the interaction coefficient.10-12For polymer concentrations used here, we shall only use first-order concentration terms, so that we are required to evaluate the constants (a2),,(aJ2,and a+. It is easily shown that for binary and ternary systems
Comper et al. we assume that it behaves identically with the unlabeled material so that (see Appendix 2) f2.1
(27)
=hl
and we also assume the reciprocal relationship7,*
= md12 (28) which will strictly apply only for dilute solutions (see Appendix 1 eq A3 and A4). mdzl
and for a ternary system
-
(aw
Osmotic Pressure. Calculation of ( a J i may be made from osmotic pressure (n) studies on a binary system since
where M3 is the molecular weight of water. For a ternary system, calculation of a+ from osmotic pressure measurements may be made through the following equation, which is obtained through the use of eq 16 and 17 and the Gibbs-Duhem equation:
nv3
- = %(ml RT
1000
(02) + m2 + -m12 2
1
(a2)z + -m? 2
+ a+mlmz
The corresponding equation for n in terms of concentration C in mass/volume units is given as a function of the standard virial coefficient form"
Phase Separation. Calculation of a+ may be made by analysis of the critical point of the phase diagram of a mixture of components 1 and 2 through the expressions of Edmond and Ogston12
where mlCfitand qcrit are the molalities at the critical point. Dynamic Parameters. Frictional Coefficients. For a ternary system there are three independent frictional coefficients, namely, fl3, f23, and fZ1, that must be determined. Evaluation of the frictional coefficients describing solutesolvent interactions, i.e.,fi3 and f23 may be made from mutual diffusion experiments of binary d i f f ~ s i o nthrough '~ the equation
where Di is the mutual diffusion coefficient of i. Evaluation of f21 has been made from the intradiffusion of tritium-labeled PVP in dextran solutions. The intradiffusion coefficient, Di* of component i is given by the equation13 Di* = R T / n j
(for all i #
J)
(26)
In using labeled material (designated by the superscript asterisk) (1 1) See, for example, H. Tompa, "Polymer Solutions", Butterworths, London, 1956. ( 1 2) Thermodynamic interactions in systems polymer 1/polymer 2/solvent have received wide attention. Early quantitative discussions are found in P. G. Flory, 'Principles in Polymer Chemistry", Cornell University Press, Ithaca, NY, 1953. Thermodynamic experiments on similar systems as used in this study are found in E. Edmond and A. G. Ogston, Biochem. J., 109, 569 (1968). (13) W. D. Comper, M.-P. Van Damme, and B. N. Preston, J . Chem. SOC., Faraday Trans. I , 78, 3369 (1982).
-
Experimental Section Materials. Dextrans T10 lo4), T70 (Mw 7 X lo4), T150 1.5 X lo5), and T500 (aw 5 X lo5)were from Pharmacia Fine Chemicals, Uppsala, Sweden. These dextrans have undergone extensive investigations of their physicochemical properties. l 4 Poly(vinylpyrro1idone) (PVP-360) (Lot No. 8 1C-2090) was from Sigma Chemical Co., St. Louis, MO. Tritium labeling of dextran and PVP was performed by the method described previo~sly.'~ Methods. The preparation of polymer solutions in water has been described el~ewhere.~ Routine measurement of polymer concentration was achieved by several methods including (i) the counting of tritium labeled samples of known specific activity, (ii) measurement of dextran concentration by optical rotation [a] at 578 nm where [a]= 2 X deg m2 kg-I, and (iii) measurement of PVP concentration by absorbance at 227 nm where the extinction coefficient is 2.0 X mz kg-I. Osmotic pressures were measured on a Melabs recording osmometer, Model CSM-2 (Wescan Instruments Inc., California), using Selectron filter, type B19 (Schleicher and Schull, Dussel, West Germany), or Sartorius membranes, type 11739 or type 11539 (Sartorius, Gottingen, FRG). All measurements were made at 25 OC. The strain gauge was calibrated with a 10 cm head of distilled water. Phase separation experiments were carried out following the procedures described by Edmond and Ogston.I2 Radioactivity counting of 3H was performed by liquid scintillation counting (Packard Tri-Carb Spectrometer, Model 3003, and LKB-Wallac 1215 Rack p scintillation counter). The activity of a 1 .O-cm3 aqueous sample was determined by using 4.0 cm3 of a scintillation mixture described by Fox.15 Care was taken that polymer samples were suitably diluted to avoid precipitation in the scintillation fluid. All samples were made to contain equal amounts of polymer and thus were equally quenched. The samples were stored in a dark cold room overnight prior to analysis. Mutual diffusion analysis was performed in the analytical ultracentrifuge by using Schlieren optics and the employment of the width:half-height method of ana1y~is.l~ Measurement of the transport of labeled PVP in dextran solutions was made in a newly developed transport cell which has been described in detail elsewhere.I6 All transport measurements were carried out at 20 f 0.5 OC. For each diffusion coefficient at least ten fluxes, corresponding to varying times, were measured. As mentioned previously, estimates of the ternary diffusion coefficients are to be performed through the use of measured values of all the necessary thermodynamic and binary frictional coefficient data together with theory outlined in previous sections. Experimental measurement of the ternary diffusion coefficients is severely restricted by (1) the wide range of initial component distributions with which structured flows may occur in dextran: PVP solutions and (2) the anticipated concentration dependence of the polymer diffusion coefficients as seen in binary systems, and therefore the requirement to perform diffusion experiments at concentration distributions similar to conditions which give rise to structured flows. We have attempted to measure these coefficients in the Sundelof transport cell by placing a mixture of solute components below the boundary with an overlay of solvent; under
(nw
-
-
(14) B. N . Preston, W. D. Comper, A. E. Hughes, I. Snook, and W. Van Megen, J . Chem. SOC.,Faraday Trans. I , 78, 1209 (1982). (15) S. Fox, Inf. J . Appl. Radiat. Isof., 19, 717 (1968). (16) L.-0. Sundelof, Anal. Biochern., 127,282 (1982); T. C. Laurent, B. N. Preston, L.-0.Sundelof, and M.-P. Van Damme, ibid., 127, 287 (1982).
The Journal of Physical Chemistry, Vol. 88, No. 6,1984 1071
Multicomponent Transport
TABLE I: Physicochemical Properties of Polymers Calculated from Osmotic Pressure Measurements 10-4a,/ kg of
polymer dextran T70b dextran T 15 Ob
*/
dextran T5 00' PVP-36 0'
2t
M,/105
solvent mol-' a
1 04A ,/mol 8.2 cm3 d
0.44 1.09 2.98 3.32
0.167 1.27 6.93 14.38
4.31 5.34 3.89 6.53
a Calculation ofa, from eq 20 employed the value of%, to convert concentrations from a mass/volume scale to a molal scale. @, anda, values were calculated from osmotic pressures on solutions ranging in concentration up to 30 kg m P . Calculated from datagiven in Figure 1. Calculated from eq 22.
1 -
TABLE 11: Osmotic Pressure Data for Mixtures of PVP-360 and Dextran
1 5 Polymer
10 Concentrotion
15
,/ k g
10-3n/
20
C,/kg m-3
m.3
Figure 1. The variation of the reduced osmotic pressure (II/c) with polymer concentration (C/kgm-3) for dextran T500 ( 0 )and PVP-360
these conditions structured flows do not develop. The apparent diffusion coefficient of a solute species i is given by the equation
Reiterating eq 1 and 2 we also have
(17) L. E. Miller and F. A. Hamm, J . Phys. Chem., 57, 110 (1953).
10-4~tjkgof solvent mol-' a
5.02 5.11 5.01 10.33 10.33 10.87 15.69 15.59 15.69
4.78 8.39 21.02 8.02 10.98 18.38 10.58 14.55 22.50
10.9 13.1 19.2 12.9 12.3 14.7 12.2 13.2 15.0
Dextran T70-PVP-360 6.82 13.27 24.5 1 6.63 13.67 19.65 8.23 15.32 18.85
which on rearranging with eq 29 gives
Results Thermodynamic Parameters. Osmotic Pressure. Osmotic pressures of binary solutions of PVP-360 and dextran T500 in water have been evaluated over a concentration range of 5-20 kg m-3. The reduced osmotic pressure (II/C) was found to be linear with concentration (Figure 1). This demonstrates that nonideality of these polymer solutions can be satisfactorily described by second-order concentration terms as embodied in eq 20. The values of the number average molecular weight, A?,,, and the nonideality coefficient a2calculated from data in Figure 1 are summarized in Table I. The values for dextrans T70 and T150 are also given. The values of a2 for the dextrans are in good agreement with those published p r e v i ~ u s l y . ' ~The J ~ a2 values increase with increasing molecular weight of the dextran. On the other hand, values of A2 remain essentially molecular weight indepedent. No published data on a2 for PVP-360 could be found for comparative purposes. Results for the total osmotic pressure of mixtures of PVP360-dextran are given in Table I1 together with the values of a+ calculated from eq 21. Dextrans T70 and T500 were studied; three concentrations of PVP were used, namely, approximately 5, 10, and 15 kg m-3, with a range of 8-23 kg m-3 in dextran concen-
dyn
Dextran T5 00-PVP-360 12.01 18.09 32.30 10.68 16.55 22.57 8.47 13.09 20.14
(a.
Therefore an estimate of Dij(in mole units) may be obtained from eq 30 providing certain gross assumptions are made concerning the values of (Dji),and (am,/em,),. These assumptions are discussed in the following section. Solution densities were calculated by using partial specific volumes for dextran of 6 X lo-" m3 kg-' (ref 12 and 13) and for PVP of 8.02 X m3 kg-l (ref 17). All computations were done on a Vax Model II/780 computer (Digital Equipment Corporation, USA).
C,/kg m-3
a
4.81 4.80 4.77 9.92 9.91 9.92 13.45 13.45 13.45
5.97 12.05 25.68 8.55 16.12 25.39 13.23 22.30 27.60
1.68 2.01 2.48 1.83 1.82 2.42 2.11 2.10 2.18
Calculated from eq 2 1 with the use cf data in Table 1.
TABLE 111: Interaction Coefficients of PVP-360 with Dextran dextran T5 00 T70 T10
10-4a+/kg solvent mol-' 13.7a 13.3b 9.5c 2.10a 0.2ae 0.4gb
104~+/moi
mol g-' 13.8 13.4 9.6 14.4
Mean values calculated from osmotic pressure data in Table 11. phase diagram by using eq 23 and 24, respectively. Data from Figure 2 and ref 3. Calculated from eq 22. e Calculated by assuming a molecular weight independentAt value of 1 4 . 1 X mol g-* C ~ T - ~ . a
b , c Calculated from
trations. For each series of measurements made on a PVP-dextran mixture, a relatively constant value of a+ was obtained. Variations in the magnitude of a+ were assumed to be due to experimental error. The mean values of a+ are given in Table 111. The value of a+ increases with increasing molecular weight of the dextran, whereas the value of A+ remains relatively constant. Mixtures of dextran T10-PVP-360 could not be studied in the membrane osmometer as dextran T10, being relatively low in molecular weight, was permeable to the membranes used. Phase Diagram. Mixtures of PVP-360 and dextran T500 separate into two phases at the concentrations shown in the phase diagram in Figure 2. For ternary transport studies where mixtures of PVP-360 at 5 kg m-3 and dextran T500 are employed, the dextran concentration may be as high as -60 kg m-3 before phase separation occurs.
11072 The Journal of Physical Chemistry, Vol. 88, No. 6, 1984
50 Dextron
100
Concentration
/ kg
rn-3
Figure 2. Phase diagram of the PVP-360-dextran TSOO system at 20 & 0.5 O C . The solution represented by the open circle consisted of one phase only. 1
7L 5. N
E
0
A 4-
\
a
= I
:z
50
0
I
100
D e x t r a n Concentration
150 kg m-3
Figure 3. Variation of the mutual diffusion coefficient with polymer concentration for dextran T500 measured by the use of Schlieren optics in an analytical ultracentrifuge.
The critical concentrations of PVP-360 and dextran are 18.5 and 38.0 kg m-3, respectively. The evaluation of a+ through eq 23 i s in excellent agreement with the a+ value from osmotic pressure. On the other hand, the at value obtained from eq 24 is somewhat lower. These differences, while subject to inherent experimental error in the measurement of mlCrItand (a2),,are amplified through the use of polymer fractions which are polydisperse with respect to the molecular weight. This is particularly = 2.98 X the case for the dextran T500 fraction which has lo5 (Table I) and Gw 5.0 X lo5 (manufacturer's specifications). The a+ value of 4.9 X lo3 kg of solvent mol-' was obtained from the phase separation experiments of T10-PVP-360 mixtures3 (Table 111). Binary Frictional Coefficients. Evaluation of polymer-water frictional coefficients, namely, f 1 3 and f 2 3 ,have been made from graphically smoothed experimental data of the concentration dependence of the mutual diffusion coefficient, together with the use of eq 26 and the nonideality coefficients in Table I. In general, the polymer--water frictional coefficients are strongly concentration dependent and increase with increasing polymer concentrati~n.'~ The mutual diffusion data for dextran T500 is shown in Figure 3. Mutual diffusion measurements of the other dextrans have been studied extensively and published e1se~here.I~ The mutual diffusion coefficient of PVP-360 over a boundary separating solutions of 5 kg m-3 of PVP and solvent has been determined by two different methods. First, by a boundary relaxation method in an analytical ultracentrifuge using Schlieren optics which gave a value for the mutual diffusion coefficient (at a mean PVP concentration of 2.5 kg m-)) of D2(2.5) = 1.1 X lo-'' m2 s.'. The second method employed the use of [3H]PVP-360 transport in the Sundelof cell; this method gave a higher value of D2*(2.5) = 2.3 X lo-'' m2 S K I . This latter value probably represents the result of preferential tritium labeling of low molecular weight species in PVP together with the possibility of some degradation that occurs during the labeling procedure.
-
Comper et al. Evaluation of the polymer-polymer frictional coefficient, namely f i l , has been made through the study of the intradiffusion of [3H]PVP-360at 5 kg m-3 in either dextran T500 or dextran T10 solutions of varying concentration. Ideally, for intradiffusion experiments, all gradients in chemical potential should be zero. However, the formation of a mechanically stable free-liquid boundary under such a condition is virtually impossible. Therefore, in order to stabilize the system, we have employed a small concentration gradient of the dextran; this is superimposed upon the uniform concentration of 5 kg mW3of PVP-360. The variation in D2* for PYP in dextran T500 and T10 solutions is shown in Figure 4, a and b, respectively. We have found essentially no difference in the magnitude of Dz* in dextran T500 solutions with gradients of 1 or 0.5 kg m-3 whereas the existence of a 5 kg m-3 dextran gradient appears to increase the magnitude of D2* at low dextran concentrations. For dextran T10 solutions, the magnitude of D2* appeared to be independent of dextran concentration gradients of 5 to 1 kg m-3. In general, the values of D2* decrease continuously with dextran concentration in a manner similar to intradiffusion measurements of [3H]dextran in dextran.I4 The values of f2' have been evaluated through the use of graphically smoothed D2* values shown in Figure 4, the normalization of these values by multiplication by the ratio of D2(2.5)/D2*(2.5) to account for the differences in mobility of the trace material to the unlabeled, bulk material, and the use of eq 26. This calculation of f i l has assumed that f23 remains constant with varying dextran concentration. We also assume in further discussion that the magnitude of f i l will be independent of PVP concentrations less than 5 kg m-3. To evaluatefI2, we employed the reciprocal relationship as embodied in eq 28 and thefil values. Experimental evaluation of f i 2 , by intradiffusion of, say [3H]dextran in 5 kg m-3 PVP solutions, is considerably less accurate and therefore was not attempted, because of the dominance of the fl.l term in eq 26, particularly at high dextran concentration~.'~*'~ Another problem that was envisaged with the intradiffusion measurements was the possibility of convective instabilities occurring due to the propensity of dextran-PVP solutions to form structured flows. However, in the case of [3H]PVP-360 intradiffusion, with low dextran concentration gradients no evidence for nondiffusional type behavior was found, Le., all measurements exhibited normal diffusion kinetics and no rapid polymer transport was measured. Additionally, by scaling the intradiffusion data according to the relationship D2* a
C y
we find y = 0.26 for dextran T I 0 and 0.17 for dextran T500. These values are unaccountably lower than those for the intradiffusion of dextran in dextran where y = 0.7-0.8 was ~ b t a i n e d ' ~ (note that the dextrans studied had molecular weights 100 kg m-3,
C , / kg
- -
(21) See, for example, P. G . De Gennes, Nature (London), 282, 367 (1 969). (22) T. J. McDougall and J. S . Turner, Nurure (London),299,812 (1982). (23) J. S.Turner, Ann. Reu. Fluid Mech., 6, 37 (1974). (24) H. E. Huppert and J. S.Turner, J . Fluid Mech., 106, 299 (1981).
4
19 70-75b
tn-3
ibi
I I
\
\
10
20
30
C , / kg tn.3
Figure 7. Comparison of inequalities (Y)designed to predict density inversion according to eq 33 (graph 1) and finger instability which is predicted to occur when Y = (1 - ijz)d,,/(l - ij1)dl2< 1 (graph 2) (from
ref 22) for (a) dextran 500-PVP-360 and (b) dextran T70-PVP-360.
to describe “hydrodynamic” instabilities they have arrived at a limiting condition for fingering systems with component distributions as studied in this investigation. Their relationship is compared in Figure 7 with inequality 33 (which is based on hydrostatic instability associated with a density inversion) for two systems, namely, dextran T500-PVP-360 and dextran T70PVP-360. In both systems the hydrodynamic instability is predicted to occur (when Y < 1) at lower dextran concentrations than
1076 The Journal of Physical Chemistry, Vola88, No. 6, 1984
predicted for the hydrostatic instability. It is apparent that the dextran concentration required for rapid polymer transport and structured flow formation correlates with the hydrostatic instability predicted for the dextran T70 and T500 systems (see Table VI and Figure 7). The variable predictions associated with dextran T10 result from the difficulty in obtaining an accurate a+ value for this system. Whether very real differences exist between the C+values and those predicted from hydrodynamic stability theory have to be considered in light of the errors discussed above, particularly in the value of DI2. The parameter is dominated by errors in measurement of a+ and M2 and not severely by the polydispersity of component 1 as reflected in an error in ml (since a+ A+MIM2where A+ is essentially independent of molecular weight).
-
Note Added in Proof. A recent publication by McDougal125 has given a broad, fluid dynamic approach to describing hydrodynamic instabilities that may occur in double diffusive convection and in systems with solute distributions similar to those studied in this paper. Differences exist in the nature of the initial concentration gradients of the gradient forming material used in this study as compared to those employed by McDougall. Acknowledgment. This work was supported by grants from the Australian Research Grants Committee and a Research Fellowship from Monash University to W.D.C. We thank Dr. W. H. Murphy for performing some of the intradiffusion experiments of PVP-360 in dextran T10. We acknowledge the excellent assistance of Ron Maxwell with the computer programing. Appendix 1 Relationship between Phenomenological Coefficients and Frictional Coefficients.
Comper et al.
I[ z ]
where
e=
1
+
x
Note that derivation of eq Al-A5 did not employ the reciprocal relati~nship,~** namely
mitj = mhi It is seen from eq A3 and A4 that this relationship only holds in dilute solutions.
Appendix 2 We set out here to establish the basis for the relationship defined in eq 27 that f2.l
(27)
=f21
In an intradiffusion experiment, ideally with a uniform distribution of the different (not trace labeled forms) components throughout, i.e.
aml am3 am2 am2* _ -- +-=o ax
ax
ax
(A6)
ax
then from eq 15
-ap2/ax = f i l U 2
+ f ~ - v~2 4 +4h3v2~ ~
-ap2*/ax = f 2 * I ~+2 f. 2 . 2 ( ~ 2-*v2)
+ f2*3~z*
since v1 = v3 = 0. Furthermore since ( ~ 2 ~= )( ~~ 2 ~ ) and ~; (a3)2* then from eq 16 and 17 it can be shown that
(~7) (~8)
(a3)2
=
Therefore, by solving the simultaneous eq A7 and A8 for v2 and v2., using the relationship that the total flux J2 J2. = 0, Le.
+
m2v2+ m28v2.= 0
(A10)
and substitution of eq A9 into A10 gives the following equality
m2m2*Cf2*1+ f2'2 + f 2 . 3 ) - f22*m22= m2m2*(f21+ f 2 2 *
[ + ""1 1
m3v3
*>
+ *[h3( m3v3 1 + m3 v3 + f 2 1 ] } / Q
+ f 2 3 ) - m2*2f2*2(A1 1)
By employing the reciprocal relationship (which only applies for dilute solutions, see Appendix 1)
mf22* = m2af2.2
('412)
then
h*1+ f 2 . 3
= f21 +f23
('413)
so by correspondence eq 27 follows. Registry No. Dextran, 9004-54-0; poly(vinylpyrro1idone) (homopolymer), 9003-39-8.
( 2 5 ) T. J. McDougall, J . Fluid Mech., 126, 379 (1982).
Supplementary Material Available: Two figures containing analysis of the dextran concentration required for rapid transport in PVP-360-dextran T70 and PVP-360-dextran T500 systems (2 pages). Ordering information is given on any current masthead page.
