J. Phys. Chem. 1983,87,667-673
Acknowledgment. This project was supported by the Australian Research Grants Committee (Grant Nos. D68/16878, D273/14137 and DS79/15252). We acknowledge the assistance of Geoffrey Wilson with analytical ultracentrifugation and Dr. W. H. Murphy for
887
supplying data of PVP-360 transport in dextran T500 solutions. Registry No. PW, 9003-39-8;PEG, 25322-68-3;PVA, 900289-5; dextran, 9004-54-0.
Kinetics of Multicomponent Transport by Structured Flow in Polymer Solutions. 4. Relationships between the Formation of Structured Flows and Kinetics of Polymer Transport W. D. Comper,' B. N. Preston, T. C. Laurent,' G. J. Checkley, and W. H. Murphy Blochemistry Diepartment, M n a s h University, Clayton, 3 168, Victoria, Australia (Received June 1, 1982)
Several different systems which generate structured flows of varying rates and morphology have been investigated. The comparison of spatiotemporalaverage measurements associated with labeled-polymer transport in these systems to topographical parameters associated with structured flows, observed by tagging the polymers with blue dye, reveals that there is little relationship between the two forms of measurement. We have shown, however, that transport of the blue-dye-labeled polymer is identical with its tritiated counterpart. The various aspects of the etiology of the process to yield structured flows are discussed in relation to linear irreversible thermodynamics.
Introduction It has been p r e v i ~ u s l y ~ shown - ~ that ternary systems consisting of a uniform concentration of dextran with an imposed concentration gradient of poly(vinylpyrro1idone) (PVP) exhibit two striking features, which are as follows: (i) the transport of either radioactive-labeled dextran or PVP in the system is extremely rapid as compared to its diffusional transport in binary systems, and (ii) this rapid polymer transport is accompanied by the formation of visible, coherent structures in the solution as seen by labeling either of the polymers with blue dye. These phenomena occur when the net macroscopic density decreases upward or even when the lower solution is supplemented with either low-molecular-weight solutes or dense solvents such as D20 to afford further macroscopic density stabilization. Certainly, further examination of structured flow phenomena, as performed in this study, has been stimulated by the probable importance of this type of phenomenum in transport and organization in a number of biological systems such as the polysaccharide compartments of the extracellular matrix of connective tissue6.' and the axoplasm of nerve axon.8 A full understanding of this phe(1)On sabbatical leave from the Department of Medical and Physiological Chemistry, Biomedicum, Uppsala University, Uppsala, Sweden. (2) B. N. Preston, T. C. Laurent, W. D. Comper, and G. J. Checkley, Nature (London),287,499 (1980). (3) T. C. Laurent, B. N. Preston, W. D. Comper, G. J. Checkley, K. Edsman, and L.-0. Sundeldf, J. Phys. Chem., first of three preceding articles in this issue (paper 1). (4) B. N. Preston, W. D. Comper, T. C. Laurent, G. J. Checkley, and R. G. Kitchen, J. Phys. Chem., second of three preceeding articles in this issue (paper 2). (5) B. N. Preston, W. D. Comper, G. J. Checkley, and R. G. Kitchen, J.Phys. Chem.,third of three preceding articles in this issue (paper 3). (6)B. N. Preston, T. C. Laurent, and W. D. Comper in 'Glycoeaminoglycan Assemblies in the Extracellular Matrix", D. A. Rees and S. Amott, Eds., Humana Press, in press. (7) W. D. Comper and B. N. Preston, Biochem. Int., 5, 557 (1981).
nomenon has not been achieved, so as an initial approach we have been performing qualitative investigations in the area. Similar visible structures in solution have been observed in other systems including the double-diffusive convection phenomena involving the interdiffusion of low-molecular-weight substances9J0and droplet sedimentation involving the interdiffusion of polymer and salt solutions."J2 The similarity of these visible hydrodynamic phenomena would give cause to the tentative speculation that there are similarities in their underlying molecular processes. These are complex systems in which a full theoretical understanding has not been achieved. It is not the purpose of this paper to make a comparative study of these different systems. What is apparent, however, is a complete lack of information concerning the relationship between the transport of molecular components in the visible structures and the formation and transport of the structures themselves. It is the purpose of this study to make an analysis of this relationship, using the PVP-dextran and modified systems, so that changes introduced into the type and morphology of the structures may be associated with changes in the kinetic behavior of measured molecular transport. A study is made of the features of labeled-PVP transport in dextran solutions compared to the structured flows formed in corresponding systems for three different conditions: (i) zero concentration gradient of dextran, (ii) zero concentration gradient of dextran together with stabilization of the macroscopic density gradient with NaC1, and (iii) stabilization of the macroscopic density gradient ~~
(8)W. D. Comper, B. N. Preston, and L. Austin, Neurochem. Res., submitted for publication. (9)J. S. Turner, Deep-Sea Res., 14, 599 (1967). (10)J. S. Turner, J . Geophys. Res., 83 2887 (1978). (11)P. Nason, V. Schumaker, B. Halsall, and J. Schwedes, Biopolymers, 7,241 (1969). (12)W.D. Comper and B. N. Preston, J . Colloid. Interface Sci., submitted for publication.
QQ22-3654/83/2087-Q667~Q~ .5Q/Q 0 1983 American Chemical Society
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Comper et al.
with a concentration gradient in dextran. 0.5,-
Experimental Section Materials. Sorbitol (Lot No. 6651330) was obtained from BDH Chemicals Ltd. (Poole, England). Dextran T10 (A&, 10000) (Lot No. FC-16026) was from Pharmacia Fine Chemicals AB (Uppsala, Sweden). Poly(viny1pyrrolidone) (PVP-360, M,, 303000, Lot No. 81C-2090) was from Sigma Chemical Co. (St. Louis, MO). Remazol brilliant blue (Lot No. 800494) was from Calbiochem (San Diego, CA). Methods. The tritium labeling and transport measurements of tritium-labeled polymers in the Sundelof diffusion cell have been described previ~usly.~,~ The covalent attachment of remazol brilliant blue to PVP was performed by the method described by Preston et al.2 The blue PVP had an extinction coefficient E1”/060(h, = 107 m2 kg-l. Visualization and Recording of Structured Flows. Still Photographs. Free liquid boundaries between two solutions were formed in the center of a vertical glass Tiselius cell (6.0 cm long, 2 cm wide, and 0.2 cm in depth). The operation was performed manually by gently placing the top solution, dispensed from a syringe through a fine plastic cannula, onto the bottom solution. The solution was overlaid with toluene to prevent evaporation. The experiments were performed in an air-conditioned room at 20 f 0.5 “C. The boundary was photographed from the side of the Tiselius cell a t regular intervals of time. A Minolta SRTlOl camera (Minolta, Japan) equipped with a 100-mm (bellows) Rotkor lens and Wratten No. 8 equivalent filter was used with a Sunpak autozoom 5000 electronic flash (Sunpak, Japan) and Ilford fan F film developed in Kodak DK50. Video Recording. The free liquid boundaries were formed in a vertical quartz glass cuvette (4.5 cm high, 0.1 cm in depth, and 1.0 cm wide) in a manner similar to that described above. A National video camera (WV-l350A, National, Japan) equipped with a Cosmicar TV lens (S-8.5 mm 1:1.5) was used to monitor structured flows. Recording was performed on a National time lapse recorder (NV8030),normally operated on the 80-h mode so that normal speed was decreased by -8OX. Recorded images were then analyzed on a monitor by tracing and calibrating; we have found that the use of the glass cell of 1-mm depth tends to restrict the structures that form to approximately a monolayer in depth. This simplifies analysis of the image so that superimposition of images with increasing depth is minimized.
-
04
-
Results and Discussion System with Zero Concentration Gradient of Dextran. In order to establish some validity in the comparison of the molecular transport of [3H]PVP in dextran to the movement of blue-labeled PVP in visible structured flows, we have measured the transport of both labeled PVP’s in dextran under identical conditions in the Sundelof cell. The standard PVP-dextran transport system4 was employed for these measurements; i.e., a Dextran T10 solution of concentration 135 kg m-3 was present throughout and with PVP-360 at 5 kg m-3 placed in the lower chamber of the cell. We have found that the blue-labeled PVP mimics the transport of its tritium-labeled counterpart in a very close manner indeed, as measured over an extended time interval (Figure 1). The linear rate of blue PVP transport is 1.2 mm h-‘ as compared to [3H]PVPtransport of 1.4 mm h-? The equivalence of the molecular transport behavior of blue PVP and [3H]PVP as spatiotemporal-averaged
0.3 0.2
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Flgure 1. Transport of bluedye-labeled PVP-360 over a boundary formed by layering 135 kg m3 Dextran T10 over a solution containing 5 kg m-3 blue-labeled PVP-360, 135 kg m3 Dextran T10. The bluelabeled PVP was monitored at an optical density of 600 nm. Q is the amount transported over a boundary of surface area A and C ois the initial concentration in the lower compartment. For comparison, the transport of [3H]PVP in the same system published previouslf is included (- -).
-
Flgtm 2. lime evolution of the macroscopic distribution of blue-labeled PVP at the boundary between solutions described in Figure 1. The solution height above and below the boundary was made to approximately 1 cm, which is approximately the same as in the Sundelof cell. The photographs were taken in order of left to right, top to bottom at times (h) of 0.08, 0.17, 0.32,0.5, 1.0, 2.0,3.0, and 4.0.
measurements in the Sundelof diffusion cell offers quite a different picture as compared with the direct observation of events occurring in a cell of approximately similar dimensions. We have photographed the transport of blue PVP-360 in the dextran system in the Tiselius electrophoresis cell over times comparable to that for molecular transport measurements in the Sundelof cell. The solution height above and below the boundary was made approximately 1cm,which is approximately the same as that used in the Sundelof cell. The time evolution of the macroscopic distribution of blue-labeled PVP is shown in Figure 2. A number of features of this system are to be noted: (i) the structures move in an irregular and tortuous manner away from the boundary; (ii) there is a marked heterogeneity in the rates of movement of various structures as the heads
The Journal of Physical Chemistry, Vol. 87, No. 4, 1983 669
Multicomponent Transport In Polymer Solutions
o0 s4
t
0
0
_--
5 Time
/
10 sec
Figure 3. Transport of various trace components Including [3H]PVP (O), and blue-labeled PVP (0)over a boundary formed by layering 135 kg m4 Dextran T10 over a solution containing 5 kg m4 PVP-360, 135 kg ma Dextran T10,2 mol dm3 NaCI, and the trace component whose transport is to be measured. For comparison, the transport of [p]PVP in the same system but without 2 mol dm-3 NaCl is shown (---).
of some structures have traversed the 1-cm column height in at least 30 min, whereas others have traversed only a small distance from the boundary; (iii) at distances away from the initial boundary, more random, turbulent behavior is noted for the rapidly moving structures, but this region eventually transforms into a more regular, collimated system after several hours; (iv) the structured flows noticeably change their morphology in their initial movement as is particularly noticeable with the enlargement of the head region of this structure while moving away from the boundary; and (v) events occurring above the boundary appear to be mirror-imaged below the boundary. It is difficult to quantitate the movements of these particular structures. A further feature of structured flows generated in systems with zero dextran concentration gradients is the correlation whereby the absence of structured flows and anomalous transport below a Dextran T10 concentration of approximately 40 kg m-3 has been noted whereas above the critical dextran concentration5 structured flows are observed. Systems with Zero Dextran Concentration Gradient and with Supplementary Macroscopic Density Stabilization afforded by NaC1. We established in the previous section the difficulty in finding quantiative relationships between structured flows and molecular transport. This necessitated modification of the system further, with the object of facilitating this type of analysis. We have previously noted that attempts to stabilize the boundary of the PVP-dextran system with low-molecular-weight solutes or D20 did not produce stabilization in terms of reducing the enhanced PVP transport rate in the system, but they did alter the morphology of the structured flows so as to produce a more uniform distribution of structures. A detailed analysis of such a system is presented here. In this case, we have arbitrarily chosen 2 mol dm-3 NaCl as an inert, low-molecular-weight density stabilizer. The kinetics of labeled-polymer transport in the Sundeliif cell of [3H]PVP and blue PVP for the system with identical polymer concentrations as studied above for the PVP-dextran system without NaCl is shown in Figure 3. The comparison of these kinetic data with those of polymer transport in the absence of the density stabilizer (Figure 1and ref 2 and 3) demonstrates that the rates were identical, i.e., 1.4 mm h-l. Additionally, it is noteworthy that the blue-labeled PVP moves in a fashion similar to [3H]PVP. The identity between the PVP-dextran systems in the presence and absence of NaCl was further established by analysis of [3H]PVP distribution in the vertical
TOP
Bottom
Flgure 4. Time distribution ofJ3H]PVP-360 associated with a boundary formed by layering 135 kg m Dextran T10 over a solution containing 5 kg m-3 [3H]PVP, 135 kg ma dextran, and 2 mol dm-3 NaCI. The boundary was formed in plastic tubes (10 X 60 mm) between solutions of 19-mm column height. Underlay fluid of tetrabromoethane was placed in the bottom of the tube. The contents of the tube was fractionated at (a) 11 and (b) 65 min by displacement with tetrabromomethane and the distributionof [3H]PVP (0)was determined. The position of the original boundary (---) is indicated. The ordinate represents the fractional percent of radioactivity in the sample, C,,as compared to that originally placed in the bottom of the tube, C,,. The abscissa represents the fractional percent distance along the tube containing solution which was approximately 38 mm in total length.
Flgure 5. Time evolution of the macroscopic distribution of blue-labeled PVP at the boundary between 135 kg m-3 dextran (top) and 5 kg m3 blue PVP, 135 kg m-3 dextran and 2 mol dm-3 NaCl (bottom). The solution height above and below the boundary was made to approxmately 1 cm, which is approximately the same as in the Sundelof cell. The photographs were taken in order to left to right, top to bottom at times (h) of 0.08 0.17, 0.25, 0.5, 1.0, 2.0, 3.0, and 4.0.
tubes at various times in Figure 4 as compared to a system without NaCl.* A remarkable feature of the NaC1-stabilized system is that, while PVP-360 transport appears identical with that of the system without NaC1, the structures formed are highly ordered and coherent (Figure 5) in marked contrast to the more irregular structures formed in the system without NaCl (Figure 2). Indeed, we have observed the development of similar highly ordered structures in any form of stabilized PVP-dextran system with low-molecular-weight solutes such as salt, and urea, sugars and dense solvents such as D20. Regularity of macroscopic behavior in the stabilized PVP-dextran system makes these systems amenable to some analysis concerning the actual development and movement of the structures. The following
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The Journal of Physical Chemistry, Vol. 87, No. 4, 1983
-
16
-
14E
1L1 2 -
.-
-t; >”
Kl8-
6-
1
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20 Time /”in
Flgure 6. Variation in the velocities of individual upward-moving stnrchired flows generated by solutions described in the length of Fgwe 5 as measured by television video monitoringof the solution in a 1-mm pathlength glass cuvette. Velocities were calculated over the interval of time in between the arrows. The first time interval corresponding to 6 min after the formation of the boundary and subsequent intervals were all 4 min in time. Approximately 50% of the upward-moving structures were recorded. At distances 3-4 mm beyond the initial boundary or after 20 min the structures are difficult to record as they tend to accelerate rapidly and also fade in color.
section represents a semiquantitative account of the events seen in these cells, as viewed and monitored by the video system in a glass cuvette of 1-mm thickness (Figure 6) and also in the Tiselius cell with still shots (Figure 5). Immediately after layering of the top solution onto the lower solution the development of a homogeneous, expanding band from the initial boundary occurs. When this zone broadens to a width approaching 0.5 mm (Figure 5 ) , the formation of small villi at each of the two satellite interfaces of the band perimeter is then observed. This behavior, which occurs in the first 5 min, is not measurably evident as a distinct process in the polymer transport kinetic studies. It is not clearly observed whether the structures that initially form at the two interfaces are related in any way, although at later times the structures, immediately above and below one another, appear to merge so that any distinguishing characteristics arising from their dichotomous origin eventually disappear. Subsequently, over a distance of 3-4 mm, fingers move essentially in the vertical plane. Measurements of the vertical displacement by the frontal “heads” of the structures demonstrate that individual structures may move at variable velocities (Figure 6) with a mean value near 10 mm h-l. (Note that these rates are considerably higher than those obtained from the Sundelof cell which were “1.4 or 2.8 mm h-’ assuming structured flows were present.) A comparison of the displacement of the upwardand downward-moving structures at a time of 14 min after the formation of the initial boundary demonstrates that the net displacement of the structures is essentially zero. Seventeen upward-moving structures were detected; they moved a total vertical displacement of 38.9 mm with an average displacement of 2.3 f 0.5 “/structure or average velocity 9.9 mm h-’/structure. Fifteen downward-moving structures were detected; they moved a total vertical displacement of 40.3 mm with an average displacement of 2.7 f 1.0 mm/structure or an average velocity of 11.6 mm h-l. The similarity in the total displacement of the upward-
Comper et al.
and downward-moving structures is in accord with the average rates of [3H]dextrantransport measured in both direction^.^,^ We have found no obvious relationship between distances moved of adjacent fingers moving in opposite directions-these distances may differ considerably, and the difference is further amplified in the PVP-dextran system without NaCl (Figure 2). Since the distances moved upward is conserved by structures moving downward then lack of immediate-neighborcooperativity would imply long-range cooperative interaction in the system. In the initial 15-20 min we find that the lateral periodicity between upward-moving structures is of the order of 0.59 f 0.01 mm. Further propagation of structures is seen to be associated with the enlargement of the head bulb and more turbulent, rapid motion of the structure after it has traversed about 3-4 mm beyond the initial boundary (see also Figure 6). It is easily seen that the most rapid structures reach the vertical limits of the system at rates approaching 20-25 mm h-’ (Figure 5). With time, the turbulent area gradually disappears and highly ordered vertical structures are propagated throughout the whole container. The structures also merge with time. There is no question that the marked change in the nature and coherency of the structures in the system containing NaCl is in some way regulated and controlled by the presence of NaC1, whose effect is manifested through the increase in density that it confers. The discrepancy between the movement of observable macroscopic structures and average kinetic molecular polymer transport is thus further exemplified in this system, as previously discussed in the system without NaCl. Stabilization of the Macroscopic Density Gradient with a Concentration Gradient i n Dextran. Another PVPdextran system which offers variability in both polymer transport kinetics and structured flow morphology as compared to systems studied is that which occurs when there exists an initial dextran concentration gradient in the system. In other words, the system is now further stabilized by the polymer component dextran. Analysis of the linear time dependence of [3H]PVPflux in such a system is shown in Figure 7. The time-dependent behavior seen for this material is similar to that observed for the PVP systems described above although its rate of transport, namely, 0.13 mm h-’ (in the first 16 h), is 1order of magnitude lower than PVP transport in a system with zero concentration gradient of dextran. Observations of the structures formed in this system over an equivalent time period are shown in Figure 8. In this case a series of relatively slow-moving spikes, highly ordered in nature, are seen to grow in an entirely uniform manner. The lateral periodicity between upward-moving structures is 0.46 f 0.02 mm, which is smaller than that observed above. The morphology of the structures appears to be quite different in this case. It is difficult to measure the frontal movement of the structures, as their extremities tend to fade. In this system, it is clearly apparent that the number of structures decreases with time, together with concomitant growth in width of structures remaining, which have lateral periodicity of -0.8-1.0 mm at 20 h. Since the observed variation in the width of fingers depends on the initial composition of the system and time, it will be important to establish what conditions actually control the size of the structures. Of particular biological interest is what happens on a more microscopic level for systems which may not show visible structures but may have microstructure, in the form of structured flows. General Discussion Comparison with Other Systems That Form Structured
Multicomponent Transport in Polymer Solutions
0.4)-
0 10
Time
/IO"
sec
Figure 7. Transport of ['H]PVP-360 over a boundary formed by layering 124.1 kg m3 Dextran T10 over a solution containing 5 kg ma [3H]PVP-360 and 146.3 kg m3 Dextran T10 as measured in the Sundeiof cell.
Figure 8. Time evoiution of the macroscopic distribution of blue-hbeled PVP at the boundary formed by solutions described in Figure 7. The solution height above and below the boundary was made to approximately 1 cm, which is approximately the same as in the Sundelof cell. The photographs were taken in order of left to right, top to bottom at times (h) of 0.08, 0.5, 1.0, 1.5, 2.0, 5.2, 16.3, and 20.0.
Flows. It is worthwhile to compare the systems studied in this paper to other multicomponent systems which generate similar boundary disturbances, such as interdiffusion of low-molecular-weight species differing in chemical compositiongJo and the layering of a polymer solution of a more dense solution containing low-molecular-weight solutel1J2(a process commonly termed droplet sedimentation). The interdiffusive phenomenon reported by Turner>lo described as double-diffusiveconvection, has been studied in some detail. The minimum requirements of double-diffusiveconvection in Turner's system are the
The Journal of Physical Chemistry, Vol. 87, No. 4, 1983 671
following: (1)The fluid must contain two or more components having different molecular diffusivities. It is the differential diffusion that produces the density differences required to drive the motion. (2) The components must make opposing contributions to the vertical density gradients. (It is assumed that the fluids are completely miscible so that surface tension effects do not arise.) Turner and his colleagues have focused their attention on systems containing low-molecular-weightcomponents such as sugar and salt. It is emphasized that these systems together with droplet-sedimentation systems may be viewed as intrinsically unstable through a distribution of material and development of density inversions, once diffusion has occurred for a finite time.13-15 In this study we have demonstrated that the requirements for rapid polymer transport and structured flow are quite different from those elaborated for Turner's system. We have substantial evidence that rapid transport (which we shall equate with structured flow formation) may merely be due to the imposed concentration gradient of the third component in a polymer-network-formingsystem. Further, we may generate rapid transport through gradients of both polymer components which are in the same direction (see experiments associated with Figures 7 and 8). These features constitute a distinctive yet general feature of the multicomponent polymer systems that we have analyzed, in which macroscopic density stabilization is afforded, in all cases, by the initial distribution of macromolecular components. We have also demonstrated that only relatively small concentration gradients of the third polymer component are required to generate rapid t r a n ~ p o r t .This ~ is consistent with the assumption that the solutions used are completely miscible and that surface tension effects would not account for the phenomena observed. The small increments in the concentration of the one of the polymer components required for rapid transport would also render it unlikely that heat of mixing (not measured) during initial transport may create local temperature gradients to perturb the system. This would also be inconsistent with the characteristic dependence of rapid transport of the flexible polymer on concentration of the dextran m a t r i ~ . External ~ perturbation of the system through temperature gradients and mechanical vibration have been effectively eliminated by using thermostated and vibration-free systems. Indeed, the rapid transport has been observed in a number of different types of transport measurements under different conditions. In contrast to the finding of Nason et a1.l1 that convective disturbance arising from the interdiffusion of polymer and low-molecular-weight solute is minimized or prevented in the presence of a centrifugal field, we have found that rapid polymer transport is enhanced with increasing g on this ~ystem.~ An interesting similarity between the polymer systems studied here and those of oscillating instabilities in gaseous multicomponent diffusion appears to exist.16J7 It is known that, when a third "solvent" gas is added in equal amounts to areas containing interdiffusing gases, then dynamic coupling may occur to give rise to density inversions. In an analogous manner, the dextran may be regarded as the pseudosolvent in our systems. The possibility of dynamic (13)H.Svenason, L.Hagdahl, and K. D. Lemer, Sci. Tools, 4,1(1957). (14)R. P. Wendt, J. Phys. Chem., 66,1740(1962). (15)V. N. Schumaker in "Advances in Biological and Medical Physics", Vol. II,J. G. Laurence and J. W. Gofman, Eds., Academic Press, New York, 1967,p 276. (16)L.Miller, T.H. Spurling,and E. A. Mason, Phys. Fluids, 10,1809 (1967). (17)L. Miller and E. A. Mason, Phys. Fluids, 9, 711 (1966).
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coupling between various components is discussed further below. Criteria for Establishing the Time Relationship of Rapid Transport. It is difficult to define the kind of motion that the polymer is undergoing during rapid transport. We have found that polymer transport measured by various techniques does not obey normal diffusional kinetic^.^-^ On the other hand, a major part of the transport process (corresponding to 6040% relaxation of the initial concentration gradient) involves a linear time r e l a t i ~ n which ~ - ~ may be viewed as a form of convective motion; we speculated previously that this type of initial linear response was consistent with the formation of structured flows. Other evidence for the validity of describing the process as convection (cf. diffusion) has been directed a t establishing the physical significance of the change from fast to slow transport in the linear time response curves. For the standard PVP-360-Dextran T10 (135 kg m-3) system with no dextran concentration gradient the transition from fast to slow transport occurs a t approximately 2.5 h and corresponds to approximately 70% relaxation of the PVP gradient (Figure 1). On the other hand, with an initial gradient in dextran concentration such as the conditions described in Figure 7 we find that the transition now occurs a t approximately 14 h with a corresponding 40-45% relaxation of the PVP concentration gradient. Thus, there is no obvious relationship associated with parameters of the linear time response curve and the PVP concentration distribution in the cell. Another approach that we have made to establish the nature of transport in the initial stages of structured flow formation has been the analysis of blue PVP transport. Spatiotemporal-averaged transport measurement in the Sundelof cell demonstrated that [3H]PVP and blue PVP behaved identically. However, a disparate relation exists between the observed motion of the structured flows and rates of molecular transport measured by space-averaged techniques. Individual structured flows may move at linear rates up to 15 mm h-’ (Figure 6) in the first 0.4 h in the PVP-dextran system, which is much faster than the 1.4 mm h-’ rate measured in the Sundelof cell. Further, observation of morphological events at times corresponding to the transition from fast to slow transport in the Sundelof cell revealed no obvious alteration in the behavior of the structures at that period of time. No marked changes were observed in the tube fractionation experiments either. In other systems in which structured flows are observed, namely, the dextran-sorbitol system,18we have observed diffusive kinetics of polymer transport rather than linear time response kinetics. Up to this stage we have little experimental evidence to justify the use of any particular type of kientic relationship by which to describe transport associated with structured flow formation. The results also suggest that the most frontal structures that are observed do not represent significant quantities of material, in a quantitative transport sense, although their surface areas appear large. It could be argued that the wide surface area boundary employed in forming the structures for photographic record bears no relation to the smaller boundary area employed in the diffusion cell. However, similar structures are observed in cells of 0.5-cm width. We suggest that various forms of transport of the PVP are occurring (in the form of structures that move at different rates, etc.) and that we only observe the most rapid phase while most of the material is still associated (18) W. D. Comper, G. J. Checkley, and B. N. Preston, Proc. Aust. Biochem. Soc. 14, 29 (1981); Comper et al., unpublished work.
Comper et al.
with local boundary regions. There is probably a good deal of “microstructure” associated with each structured flow which has not been observed. Density Stabilization. A feature of the many systems associated with rapid polymer transport reported in this paper is the perceived stability afforded by the initial distribution of components and macroscopic density gradients. In tagging the polymers with dye, we have found, however, that rapid transport is accompanied by the development of structured flows in the solution. The possible occurrence of density inhomogeneities through the redistribution of macromolecular material at the initial boundary to ultimately yield structured flows also has to be considered in relation to attempts to stabilize the boundary by the addition of low- or high-molecularweight materials in the lower solution to increase the macroscopic density gradient across the boundary. For low-molecular-weight density “stabilizers” we have found little effect on the kinetics of polymer transport in the system. We tentatively propose that this material is able to equilibrate rapidly in regions where marked concentration fluctuations of macromolecular material occur. The presence of low-molecular-weight stabilizers did have a markedly unifying effect on the size and rate of the structured flows formed, however. This again points to the heterogeneity of the process in which spatiotemporal-average measurements are not apparently affected by the presence of a stabilizer, whereas individual local structures are. The addition of macromolecular density stabilizers to the lower solution had significant effects depending on the amount and chemical nature of the material added. The addition of matrix material (i.e., dextran) led to a significant decrease in the rate of enhanced polymer transport, although even at concentration gradients of 20 kg mW3enhanced rates and also structured flows are still observed. On the other hand, increasing the concentration of PVP in the lower compartment had little effect on polymer transport in the ~ y s t e m .While ~ these effects are not presently understood, it is apparent that the macroscopic density per se has only indirect relationship to the formation of these structures, whereas the macromolecular composition of the solutions is of primary importance. Etiology of Structured Flows. To facilitate the further discussion, we could demarcate the formation of structured flows in two stages: (1)a consideration of molecular dynamics that yield instabilities at the boundary and (2) factors stabilizing the structured flows once formed. The etiology of events leading to the instabilities will form the major focus of the discussion presented here. A quantitative consideration of molecular motion immediately subsequent to the formation of the initial boundary is not yet possible, as not all the parameters have been evaluated. However, in assuming as a first approximation that unperturbed diffusion occurs in this initial period, which ultimately establishes density inversion, we shall pursue a qualitative examination of the system in terms of linear irreversible thermodynamics. We consider parameters which are envisaged to be important for systems with zero volume flow, but which may ultimately lead to unstable concentration inhomogeneities at the boundary. We utilize the linear irreversible thermodynamic treatment of Miller.19 We define component 1 = dextran; component 2 = flexible polymer, e.g., PVP; and component 3 = H,O. The theoretical treatment describes the flux J of components 1 and 2, with the initial condition that de, f ax = 0, as (19) D. G. Miller, J. Phys. Chem., 63,570 (1959).
Multlcomponent Transport in Polymer Solutions
The Journal of Physical Chemistry, Vol. 87, No. 4, 1983 673
where
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b’ = a In a2/ac2
(5) (6)
where Dij are the diffusion coefficients, L , are the phenomenological coefficients, ci is the concentration of component i, Vi is the partial molal volume of component i, and ai is the activity of component i. For convenience, we may interpret the diffusion coefficients in eq 1 and 2 as being functions of both dynamic parameters (such as the phenomenological coefficients) and equilibrium thermodynamic quantities as embodied in the activity terms. The diffusion coefficients involve cross-interaction terms in both these equilibrium and dynamic functions. It is clear that the magnitude of the terms a In C Y ~ / and ~ C ~a In a2/ac2 will be critical determining the magnitude of the diffusion coefficients in eq 3 and 4. An accurate assessment of the magnitude of these terms, in the range of concentrations studied in the transport measurements, is currently being undertaken. It is expected on the basis of the polymer vinal coefficients and theoretical evaluation of interaction coefficients that values of a In al/dc2 are significant. Therefore, from purely thermodynamic considerations we could predict that the presence of the PVP induces a chemical potential gradient in the dextran, thus causing the polysaccharide network to move. From dynamic considerations, the movement of the network may also arise from dynamic coupling of PVP flux as manifested through the phenomenological terms LZl or L12in eq 3. Experimental evaluation of these terms in situ is difficult owing to the extremely short period in which unperturbed diffusion may occur prior to the onset of rapid motion which occurs over a long time scale. Development of a local density inversion at the boundary therefore necessitates a preferential net transport of dextran as compared to PVP transport from the dextran-PVP solution into the dextran solution above it. While this type of explanation presents general features that may occur initially, we have in no way accounted for the critical dextran concentration required for a network-forming system. As we have not observed this parameter in physicochemical studies of binary systems,s*6 such a parameter may only be associated and observed in the equilibrium and dynamic interactions between PVP and dextran. Up to this stage, we have been primarily concerned with dynamic and equilibrium interactions associated with polymer mixtures undergoing unperturbed diffusion with zero volume flows. We now consider the various aspects of the system associated with the generation of volume flows in the systems studied. Obviously no macroscopic volume flow can occur in systems with a free liquid boundary. However, it is informative to consider volume
flow behavior only in microscopic volume elements. If the partial specific volumes of the polymer components depend on concentration, a volume change will generally occur during diffusion and result in a hydrodynamic or bulk flow relative to the position of the initial boundary. However, we consider that with the use of small increments in the concentration of flexible polymer used in our experiments this factor is not important in manifesting density inversions at the boundary. Neglecting the movement of the dextran matrix, we may define the total volume flow J , as J, = J2V2 + JSV, For nonzero values of J, and constant Vi (vide supra) we may define J, in terms of pressure gradients in the system20 such that
an
aP
J,=A-+A (7) ,ax pD ax where A, and ApD are phenomenological coefficients. By assuming that aP/ax = 0, one may reduce eq 7 to
J, = A,uAn
(8)
where A, represents a filtration coefficient, u the reflection coefficient, and A l l the difference in osmotic pressure. We consider that eq 7 and 8, normally defined in relation to solute transport through a membrane, may equally well apply to flexible polymer transport through the dextran “membrane” matrix. The selectivity of the solute with respect to the matrix may be defined in the modified form2‘ S=
a
c3v3 + C 2 V 2 U
When S = 1 the flexible polymer is freely permeable, whereas when S = 0 the flexible polymer is impermeable in its transport through the matrix. As discussed above, the diffusion of solutes in polymer networks is normally retarded, corresponding to a value of S between 0 and 1. The efficiency of retardation may be correlated with the formation of transient, statistical network at the critical concentration. In this case, an osmotic flow defined by eq 8 may take place. Note, however, that this type of transport has yet to be experimentallyidentified. The flow could conceivably create concentration inhomogeneities at the boundary leading to density inversion. In this context we have established, by limited studies, that density inversion of solutions at the boundary would occur if osmotic flow of water proceeded rapidly to produce conditions equivalent to a normal equilibrium dialysis e~periment.~
Acknowledgment. This project was supported by the Australian Research Grants Committee (Grant Nos. D68/16898, D273/14137, and DS79/15252). We acknowledge the expert assistance of Mr. S. V. Morton with the photography. Registry No. PVP, 9003-39-8; dextran, 9004-54-0. (20) 0. Kedem and A. Katchalsky, Biochim. Biophys. Acta, 27, 229 (1958). (21) A. G. Ogston and C. C. Michel, B o g . Biophys. Mol. Biol., 34,197 (1978).
