J . Phys. Chem. 1985,89, 128-134
128
of gm2and L, the coefficients of the two modes exp(ig,l) and exp(-ig,Z) are equal as are the eigenvector components. The coefficients of all other modes are zero. Call the unique nonzero coefficient C/2. Then x, = (C/2)xm exp(igml)
+ (C/2)xm exp(-ig,l)
= kml) ( A l l )
C X m COS
with similar equations for y, and z , . The quantity Cis an arbitrary constant that determines the amplitude of the solution. The only first-order solutions to eq 4.1-4.3 with the given boundary conditions are then (xo, yo, zo) plus terms proportional to a simple cosine function; these solutions exist only at critical container lengths, L,. Such solutions exist for the parameter values used here only when 0.61 < f I 1.82.
Kinetics of Multicomponent Transport by Structured Flow in Polymer Solutions. 7. Polysaccharide-Saccharide Systems' W. D. Comper,* M.-P. I. Van Damme, G. J. Checkley, and B. N. Preston Department of Biochemistry, Monash University, Clayton, Victoria 31 68, Australia (Received: March 14, 1984)
A quantitative examination of the properties of the ternary system of polysaccharide/saccharide/water has been performed over a wide range of concentrations and molecular weights of solutes. This has included the analysis of polymer transport and structured flow formation, together with the analysis of the excluded volume and frictional interactions that determine the ternary diffusion coefficients and density profiles across the boundary. Both the hydrostatic and hydrodynamic instability models underestimate the solute concentrations required for the flow regime. In certain cases, a correlation exists between the magnitude of a density inversion that forms through cross diffusion and the rate of polymer transport and flow formation. In other systems, notably with relatively low molecular weight solutes, more complex relationships exist which may be due to relatively large density fluctuations in the density inversion that forms. An intermediate solute concentration range corresponding to the transition of the system from a normal diffusional regime to a structured flow regime has been identified. The molecular interactions giving rise to the structured flow regime are quite general and therefore the results are expected to apply to other polydisperse polymer systems.
Introduction
Previous papers in this series have dealt with our findings of the rapid transport of poly(vinylpyrro1idone) (PVP) in concentrated dextran solutions. This rapid transport has been shown to be accompanied by the formation and movement of structured flows or macroscopic fingerlike structures that move in the gravitational field and transport high molecular weight material.24 The phenomenon has been extensively investigated over a wide range of experimental condition^.^^^ Ternary diffusion analysis has demonstrated that diffusion of PVP and dextran are strongly coupled due to large excluded volume effects between these sol u t e ~ . These ~ and other findings led to the identification of a strong correlation between the onset of a density inversion at the initial boundary, dependent upon dextran concentration (due to coupled flows), and the onset of rapid polymer transport and structured flow f ~ r m a t i o n .It ~ has been proposed that the diffusion-mediated density inversion marks the transition of the system from a diffusion regime to a flow regime. In this paper we study a ternary system of polysaccharide/ saccharide/water where the solute components are of approxi(1) Part of this work has been presented by: Comper, W. D.; Checkley, G. J.; Preston, B. N. Proc. Aust. Biochem. SOC.1981, 14, 29. (2) Preston, B. N.; Laurent, T. C.; Comper W. D.; Checkley, G. J. Nature
(London) 1980, 287, 499.
( 3 ) Comper, W. D.; Preston, B. N.; Laurent, T. C.; Checkley, G. J.; Murphy, W, H. J . Phys. Chem. 1983, 87, 667. (4) Laurent, T. C.; Preston, B. N.; Comper, W. D.; Checkley, G. J.; Edsman, K.; Sundelof, L.-0. J . Phys. Chem. 1983,87, 648. (5) Preston, B. N.; Comper, W. D.; Laurent, T. C.; Checkley, G. J.; Kitchen, R. G. J . Phys. Chem. 1983,87, 655. (6) Preston, B. N.; Comper, W. D.; Checkley, G. J.; Kitchen, R. G. J . Phys. Chem. 1983, 87, 662.' (7) Comper, W. D.; Checkley, G. J.; Preston, B. N. J . Phys. Chem. 1984, 88, 106.
0022-3654/85/2089-0128%01.50/0
mately the same chemical composition and density but vary only in molecular weight and diffusive mobility. We have previously shown, in limited studies, that such systems spontaneously generate structured flow organizations.'.s The experimental investigation has been directed a t the quantitative examination of both the diffusion and flow regimes. This includes analysis of rapid polymer transport and structured flow formation together with analysis of the excluded volume and frictional interactions that determine the ternary diffusion coefficients and density profiles across the boundary. The analysis of a ternary system consisting of solutes of the same chemical composition should provide information relevant to the behavior of other ternary mixtures such as those involved in the aggregation-disaggregation or polymerizing reactions of a solute and in enzyme-mediated reactions of polymer synthesis or breakdown. Materials
Dextrans T1 (FDR 5235) (&fw= lo3, &fn = 710), T10 (I@,,. = 1.04 X lo4, I@,,= 6 X lo3), T70 (aw = 6.95 X lo4, I@" = 3.95 X lo4), and FDR 7783 (Bw = 15.82 X 104, = 12.03 X lo4)
an
were from Pharmacia Fine Chemicals, Uppsala, Sweden. The physicochemical properties of most of these dextrans have been fully chara~terized.~Sorbitol and raffinose were from BDH Chemical Limited (Pmle, England) and sucrose was from Nakarai Chemical Ltd. (Japan). Bovine thyroglobulin (T-1001) was from Sigma Chemical Co. (St. Louis). Visking dialysis tubing (8/32-in. wide) was from Medicell International Ltd. (London, England). The dialysis tubing was treated with warm 1% (v/v) glacial acetic acid solution, washed with sodium carbonate solution, and then (8) Comper, W. D.; Preston, B. N. Biochem. h t . 1981, 3, 557. (9) Preston, B. N.; Comper, W. D.; Hughes, A. E.; Snook,I.; van Megen, W. J . Chem. SOC.,Faraday Trans. 1 1982 78, 1209.
0 1985 American Chemical Society
Multicomponent Transport
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985 129
thoroughly rinsed with distilled water before use.
of saccharides, of different molecular weight, as the gradient material. The morphology of transport in these various systems Methods has been studied with blue-dye-labeled d e ~ t r a n . ~ , ~ Labeling Procedures. The methods used for tritium labeling Evaluation of Ternary Diffusion Coefficients. Previous analysis and blue dye labeling of dextran have been described p r e ~ i o u s l y . ~ ~ ~of~ diffusion coefficients in ternary systems containing dextran and During the course of purification of [3H]FDR 7783 dextran by PVP’ demonstrated a direct correlation between the onset of gel chromatography on Sepharose 6BCL (Pharmacia, Sweden) structured flow formation and the formation of a density inversion in H20, anomalous peaks were found corresponding to apparent through cross-diffusion effects. The relationships used for calaggregates of the dextran. These aggregates were not present when culation of ternary diffusion coefficients from experimentally chromatography was performed in 0.1 mol dm-3 NaC1.9 We have measured thermodynamic and frictional interactions have already subsequently performed all experiments with dextran FDR 7783 been described.’ A similar analysis will be performed in this study. in 0.1 mol dm-3 NaCl, assuming that this solvent could be conThe isothermal diffusion in a ternary system can be completely sidered as a single-component solvent. Apparent aggregates of described by two flow equations along the x coordinate only: dextran in H 2 0 were not found for dextrans T10 or T70 and therefore all experiments with these materials were performed in H 2 0 . Tritium labeling of thyroglobulin was performed by the method described by Tack et a1.I0 The labeled product was purified on a Sepharose 6BCL column (Pharmacia, Sweden) (dimensions 1.5 In these equations, the flows (.Ii), are referred to the volume fixed X 65 cm) in 0.1 mol dmb3 NaC1. reference frame, mi is the molal concentration of i ( = 1, 2), and Preparation of Solutions. The preparation of solutions, par0 is the conversion factor of molal to molar concentration scale ticularly for the quaternary system involving dextran FDR 7783 so that aOmi/ax is the molar concentration gradient of component and 0.1 mol dm-3 NaC1, followed the procedures described prei. Full expressions for the ternary diffusion coefficients described viously by Comper et al.” Solutions were made by weight and in eq 2a and 2b, in terms of binary frictional coefficientsI5 (which their water contents adjusted to ensure there was no inverted are independent of reference frame) and thermodynamic noniconcentration gradient of NaCl for solutions used in transport deality coefficients (based on molal concentration scaleI6), have measurements. been given in ref 7. In order to compare these diffusion coeffiExperimental Protocol for Transport Measurements. All cients, it is convenient to express them on a mass unit concentration transport measurements, which are unidirectional, were performed scale, dij, where in an upright cefl1*J3with a shear-formed free-liquid boundary separating upper and lower solutions. The transport coefficient, (3) Ti, of solute i is obtained by diffusional analysis of the transport and M iis the molecular weight of i . of corresponding labeled solute across the boundary as described Evaluation of the Thermodynamic Nonideality Coefficient, by the following e q ~ a t i o n . ~ J ~ a+. Two methods have been employed to estimate a+ between Ti = ( Q / A C d 2 ( r / t ) (1) dextran and sorbitol. ( i ) Equilibrium Dialysis. The partitioning of sorbitol between Q is the total quantity of diffusing solute transported across the a dextran solution contained in the dialysis sac and the external boundary, of cross-sectional area A during the time t after the dextran free solution was determined by measuring the equilibrium formation of an initially sharp interface and Co is the initial distribution of [ 14C]sorbitolof known specific activity between concentration of labeled solute present in trace quantities. It is the two compartments. When the solutions were gently rocked assumed that the labeled and unlabeled species behave in a similar at 4 OC, equilibrium was achieved within 7-10 days. The final fashion. All transport coefficients are described with their corconcentrations of dextran and sorbitol were determined by poresponding 95% confidence limits14 unless otherwise stated. larimetry and radioactivity, respectively. Two to four sacs were Experiments were performed in an air-conditioned room mainused for each initial dextran concentration studied. N o binding tained at 20 f 0.3 OC. of [‘4C]sorbitol to the Visking tubing occurred. The dextran was used at identical concentrations (in units of The chemical potential of the permeating species (sorbitol) on g g-’ solution) in the upper and lower compartments of the cell both sides of the dialysis membrane designated by 11’ and 11’’ can and is referred to as the “matrix” (component 1). A concentration be expressed in terms of convenient algebraic expressions16which gradient of a saccharide is then superimposed upon the matrix; at constant temperature and pressure give this is referred to as the “gradient material” and is designated as component 2; the solvent is designated as component 3. To ensure (4) initial fluid density stability, the gradient material is always placed 1121’ = p20 RT(1n m; (a2)2m2/’ a+ml”) (5) in the lower solution. Studies of [’Hldextran transport have been made using dextran where (a2)2is the second virial coefficient of sorbitol and mi is FDR 7783 matrices (in the concentration range of 5-30 g kg-I) the molality of i (calculated from the number average molecular with sorbitol as the gradient materia1 (in the concentration range weight). For sorbitol the (a2)2term is negligible.” At equilibrium of 5-50 g kg-I). A number of other parameters in the poly1121 = 1121‘ saccharide/saccharide system have been varied. We have compared dextran transport in matrices of similar relative viscosity so that using eq 4 and 5 in eq 6 we obtain (and with the same gradient material) but constructed with dextrans of different molecular weight, namely, FDR 7783 at 30.0 In K = -a+ml“ (7) g kg-l and T10 at 92.5 g kg-I. Additionally, we have used a range where the partition coefficient K = m;/m2/. (ii) Frontal Gel Chromatography. The equilibrium dialysis (10) Tack, B. F.; Dean, J.; Eliat, D.; Lorenz, P. E.; Schechter, A. N. J . technique could only be applied to dextrans T70 and FDR 7783, Biof. Chem. 1980, 255, 8842. as dextran T10 passed through the Visking tubing. To obtain (11) Comper, W. D.; MacDonald, P. M.; Preston, B. N. J . Phys. Chem., estimates of a+ for low molecular weight dextrans, we have exin press. tended and developed the frontal gel chromatography technique (12) SundelBf, L.-0. Anal. Biochem. 1982, 127, 282. Laurent, T. C.; Van Damme, M.-P. Ibid. 1982, 127, 287. Preston, B. N.; Sundelof, L.-0.; of Winzor and Nichol.l* A comprehensive description of the
+
(13) Van Damme, M.-P. I.; Comper, W. D.; Preston, B. N. J . Chem. SOC., Faraday Trans. J 1982, 78, 3357. (14) Finney, D. J. ‘Statistical Methods in Biological Assays“; Griffin: London, 1971. Daniels, F. ‘Experimental Physical Chemistry”, 7th ed.; McGraw-Hill: New York, 1970.
+
+
(15) Spiegler, K. S. Trans. Faraday SOC.1958, 54, 1409. (16) Ogston, A. G. Arch. Biochem. Biophys. Suppl. 1962, I , 39. (17) Bower, V. E.; Robinson, R. A. J . Phys. Chem. 1963, 67, 1540.
Comper et al.
130 The Journal of Physical Chemistry, Vol. 89, No. 1, 1985
TABLE I: Characteristics of Polymer Unidirectional Transport Coefficient, TI, and Boundary Morphology in Various Polysaccharide Matrix/Sorbitol Gradient Svstems
concn, mg g-' sorbitol matrix gradient 5.0 5 .O
30.0 30.0 25.0 25.0 100.0 100.0
5.0 50.0 5.0 50.0
T,/10-" m2 s-' Dextran FDR 7783 2.7 (1.9-3.5) 4.4 (3.7-5.1) 6.1 f 1.4 (n = 11.3 f 0.7 (n = 3)
