J. Chem. Soc., Furuday Trans. I , 1982, 78, 3369-3378 Diffusion of Tritiated Water (HTO) in Dextran + Water Mixtures BY WAYNE D. COMPER, MARIE-PAULE I. VAN DAMME AND BARRY N. PRESTON* Biochemistry Department, Monash University, Clayton, Victoria 3 168, Australia Received 1st March, 1982 The diffusion of HTO has been measured in dextran solutions using an open-ended capillary technique and a newly developed Sundelof diffusion cell. HTO diffusion has been examined as a function of dextran concentration and molecular weight. These results, together with our previous results on the intradiffusion and mutal-diffusion coefficients of dextrans, now provide a complete set of conventional translational diffusion coefficients for both components in this binary system. Various assumptions associated with the theoretical description of polymer translational motion can now be examined.In order to understand the molecular motion ofpolymers in binary polymer + solvent systems knowledge of the relationship between the diffusional behaviour of both components is important. It is apparent, however, that in studies on aqueous polymer systems this relationship has been overlooked. Previous on the diffusion of dextran and other water-soluble polymers have been interpreted on the basis of several gross assumptions associated with the nature of the solutions and the relative movement of both polymer and water. In this study, these assumptions are tested for dextran + water mixtures. We have experimentally determined all the conventional diffusion coefficients for each component.The intradiffusion and mutual-diffusion coefficients of dextran, which have been reported ear lie^,^ are now complemented in this study with intra- diffusion measurements of HTO in dextran solutions. The relationship between the various motions of each component can now be represented and analysed in a unified and comprehensive manner. HTO diffusion is probably a good representation of molecular water motion rather than that of a special rapid H+ transfer process since HTO and H,lsO move at the same rate.4 We have studied HTO diffusion in two dextrans of different molecular weights, namely dextran TI0 (M, = 10400) and dextran FDR7783 (M, = 158200). Some measurements of the diffusion of water in dextran solutions by a proton magnetic resonance-pulse method have been reported previously5 and will be compared with values obtained in this study.EXPERIMENTAL MATERIALS Tritiated water (lot no. 1275-133, 0.25 mCi g-l)? was from - - New England Nuclear (Boston, U.S.A.). The polymer dextran samples T10 (M, = 10400; Mw/Mn = 1.68) and FDR7783 (M, = 158200; Mw/Mn = 1.32) were either supplied or kindly donated by AB Pharmacia (Uppsala, Sweden). Dextrans were labelled with tritium as described by Preston et aL3 Dextran solutions were prepared by weight in distilled water either from the dextran as supplied t 1 Ci = 3.7 x 1O1O Bq. 109 3369 FAR 783370 DIFFUSION OF kITO IN DEXTRAN+WATER or from dried samples. Conversion of polymer concentration into mass!volume units was carried out on the basis of the dry weight of the solid and the partial specific volume of the polymer.METHODS The intradiffusion coefficient of HTO in dextran solutions was measured by two techniques, namely a modified3 open-ended capillary technique of Anderson and Saddington6 and in a newly developed diffusion cell.'? Equilibrium dialysis of HTO in dextran solutions was performed at 4OC using Visking tubing (size 8/32 in, Medicell International, London).? The tubing was pre-treated with 1 % acetic acid and washed in sodium carbonate prior to use. Each dialysis bag contained dextran (at - the required concentration) and fluorescein-labelled dextran (FITC-Dx- 150, lot F- 108, M , 7 153 700 from AB Pharmacia, Sweden) at a low concentration of 1.2 kg mV3 and of known specific activity. The fluorescein-labelled dextran was used to monitor any changes in volume of the sample during dialysis.The total dextran concentration was varied up to 133 kg m-3. The dialysis bags (there were five for each dextran concentration studied) were placed in a cylinder containing HTO (ca. 70000 cpm ~ m - ~ ) . $ The solutions were gently rocked for 4 days. The final concentration of dextran in the bag was estimated by measuring the concentration of fluorescein-labelled dextran with an Hitachi spectrofluorimeter (Hitachi Ltd, Japan) using exciting light of wavelength 493 nm and reading the emission at 515 nm. The ratio of HTO inside the bag to HTO in the dialysate (as measured by radioactivity) gives the partition coefficient Kav. Radioactive counting procedures have been described elsewhere.8 HTO samples containing dextran were prepared for liquid scintillation counting with Aquasol (NEF-934 New England Nuclear, Boston, U.S.A.) (22.5% aqueous sample in Aquasol to form a stiff gel).There was no change in the counting efficiency of HTO over a wide range of dextran concentration (varied from 1 to 30 mg per counting vial). The partial specific volume of dextran T10 was determined by measuring the pyknometric density a t 20 *C of dextran solutions at concentrations of 104.8 and 217.4 kg m-3. A value of 0.60 cm3 g-l was found at both concentrations and agrees with values obtained at lower dextran concentrations by Edmond et al.9 NOMENCLATURE We designate dextran as component 1 and water as component 2 and their corresponding trace-labelled counterparts by the subscript*.We shall be dealing with three diffusion coefficients, namely the mutal-diffusion coefficient of component 1, D,, and the intradiffusion coefficients 0;' and Dl. Diffusion coefficients at infinite dilution will be further designated by the superscript O. RESULTS INTRADIFFUSION OF HTO IN DEXTRAN Fig. 1 shows the dependence of the reduced diffusion coefficient of water Di/(Di)O on the concentration of dextran. Several features of the results may be noted: (i) the reduced diffusion coefficient of water decreases with increasing dextran concentration, (ii) the magnitude of the decrease of the diffusion rate appears to be independent of the molar mass of the dextrans used in this study and (iii) the values of Dt/(Dt)" obtained from the two different methods appear to be similar. Other measurements of water diffusion in dextrans of various molecular weights5 showed property (ii) for dextrans with molecular weights up to 150000.However, the magnitude of their reduced diffusion coefficients at high dextran concentrations is considerably lower than that obtained in this study. We can offer no explanation for this difference. Note t 1 in = 2.54 x 10-2 m. $ cpm = counts per minute.W. D. COMPER, M-P. I. VAN DAMME AND B. N. PRESTON 3371 0 0.3 0 50 100 150 200 250 mean dextran concentration/kg m-3 FIG. 1.-variation of the reduced diffusion coefficient D:/(D:)O for HTO diffusion in dextran T10 (0) and dextran FDR7783 (A) as obtained from Sundelof diffusion cells. Values of D:/(D:)O obtained by the open-ended capillary technique are shown by the corresponding open symbols.that our values have been corroborated through the use of two independent methods and are also in accord with magnitude of reduced diffusion coefficients of low- molecular-weight solutes in dextran solutions.1° DISCUSSION OBSTRUCTION AND EX C LUDED-VO LU ME MODELS An approach which has proved successful in explaining the hindered transport of compact macromolecules in chain polymer solutions is the stochastic model of Ogston et This model takes into account the molecular size of the interacting species on the basis of the excluded-volume concept. The reduced diffusion coefficient is given in terms of the equivalent Stokes radius r2 of the migrating species, the effective cylindrical radius of the fibrous network molecules r1 and their effective specific volume (1) 6 so that D;/<D:>~ = A exp [ - BC!] where B = (r1+r2) Vt/r1 (2) and A is a constant close to unity and C, is the concentration of component 1 in mass/volume units.An independent estimate of B may be obtained from equilibrium partition of the migrating species between the free solution and a compartment containing dextran such that the partition coefficient (KaV) is given by1' K,, = exp [ - B2C1]. (3) Using the experimental values of K,,, the calculated values of B in eqn (1) and (3) may be compared. Linear regression analysis of the variation of In [D;f/(D:)"] against Ci yields values of A = 1.15 (kO.09) and B = 1.333 (k0.17). This value of B from diffusion data compares favourably with the values of B obtained from eqn (3) and K,, data only at dextran concentrations c ca.70 kg m-3 (fig. 2). The high error bar on B values from equilibrium partition experiments, which is due to the relatively high 109-23372 DIFFUSION OF HTO I N DEXTRAN+WATER 1 . 8 1 1 1.6 1.4 B 1 2 0.6 0.4 LL 0.15 0.20 0.25 0.30 0.35 &-/(kg rn-3)* FIG. 2.-Variation of B calculated from eqn (3) using experimental determination of K,, from equilibrium partition experiments, with C,b (0): The solid line represents the value of B obtained from linear regression analysis of HTO diffusion data in dextran using eqn (1). The filled area represents the 95% confidence limits of this line. values of Kav owing to the small effect of the dextran on water distribution for the dextran concentrations studied, precludes any further assessment of the stochastic model, particularly in estimating a sensible value of rl from eqn (2).Another treatment, which has been derived for dilute solutions, is the simple obstruction model of Wang.12 In this case, if it is assumed that exchange between bound (within the hydration layer of the polymer) and unbound water is rapid then the reduced diffusion coefficient may be expressed as where a is a geometric factor, pz is the density of water, w, is the mass fraction of the polymer and His the mass of water bound per unit mass of polymer. A calculation of H by a graphical procedure gives a best-fit value of H = 0.4 (i.e. 0.4 g of water bound per g of dextran or ca. 4.4 moles of H,O per mole of disaccharide unit).This value of H is similar to that found for DNA by Wang.13 PHENOMENOLOGICAL FRICTIONAL COEFFICIENT INTERPRETATION HTO INTERACTION WITH DEXTRAN CHAIN SEGMENTS The mutual-diffusion coefficient of dextran D, may be given a314 where pl is the chemical potential of component 1, ci and Oi are the molar concentration and partial molar volume, respectively, of component i a n d f l is the frictional coefficient15 obtained from mutual diffusion such that (6) W l -- =fE(u1-uz) ax where ui represents the velocity of component i. Note that the thermodynamic termW. D. COMPER, M-P. I. VAN DAMME A N D B. N. PRESTON 3373 14 ‘i 12 5 10 0 E 13 x 2 6 y \ 4 2 0 I I dextran concentrationlkg m-3 100 200 0 100 200 dextran concentration/kg m-3 FIG. 3.-Calculated values of fly from eqn (5) as a function of C, for (a) dextran TI0 and (b) dextran FDR7783 [values of D, were obtained from ref.(3)]. c1apl/dc1 in eqn ( 5 ) may be evaluated through a standard virial expansion (see Appendix). An estimate of the type of interaction that occurs between HTO and dextran may be made through a comparison of the flz values obtained from dextran diffusion and HTO diffusion. Evaluation off’ in eqn ( 5 ) from mutal diffusion and thermodynamic data of dextran3 is shown in fig. 3 . For both dextrans, the calculated values of fE are seen to sharply increase with dextran concentration (fig. 3 (a) and (b)]. The higher the molecular weight of the dextran the higher the magnitude of ffi. An alternative estimate of flz may be made through values of HTO diffusion in dextran.The intradiffusion coefficient of water may be described in terms of frictional coefficients, when c2* -g cz, such that14 (7) D$ = RWfZ *I +JZ *A’ Derivation of eqn (7) requires the frictional coefficients to obey the reciprocal relation and the relation c2f21 = C l f 1 2 f - 2 =f1*2 = f 1 z * The substitution of eqn (8) and (9) into eqn (7) and rearrangement yields an expression for f12 such that where ffi is defined as the f12 coefficient obtained from intradiffusion measurements through this equation as distinct fromfg in eqn (5). The~retically,f{~ andfg should be identical. Derivation of eqn (10) requires that ( D l ) O = RT/f2 * 2 (1 1) which is assumed to be constant over the concentration range of dextran studied. This assumption is probably quite reasonable in view of experimental data which suggest3374 DIFFUSION OF HTO I N DEXTRAN+WATER 0 a a a .0 0 a a I 1 I I I 0 100 200 300 dextran concentration/kg m-3 FIG. 4.-Calculated values of F{2 from eqn (12) as a function of C, for dextran T10 (0) and dextran FDR7783 (0). that polymers and polysaccharides similar to dextran have little or no effect on the molecular properties or structure of water at the polymer concentrations used in this study.16 Evaluation from eqn (10) through experimental data of D i reveals that this quantity is considerably higher (not shown) in magnitude than values o f f g evaluated from eqn ( 5 ) and given in fig. 3. However, it is of interest to convertf:, in eqn (10) into a frictional coefficient, F[2, evaluated in terms of mass/volume concentration units such that (12) The values of F,I, are found to be independent of dextran concentration and molecular weight (fig.4). The magnitude of I;;i is also seen to be considerably less than the values of ffi calculated from mutual-diffusion data [fig. 3(a) and (b)]. These results give rise to the concept that a water molecule is undergoing a frictional interaction with only a segment of the dextran molecule. This would also suggest that in order forhy [obtained from eqn (5)] to be equal tof:, [defined in eqn (lo)], the adjustable parameter should be an effective molecular weight of component 1, (M,),,,, to be embodied in the c, term in eqn (10). In this case we define I - zM2 4 2 -f12--* Ml The calculated values of (Ml)eff are given in fig.5 . It is clear that these values are dependent on dextran concentration and overall molecular weight of the dextran. The latter probably reflects the dependence of (M,),,, on the relative mobility [as embodied in eqn (6)] of the water molecule and chain segment of the dextran [which is equivalent to (A41)eff]. The data would suggest that while some form of entanglement is envisaged at higher dextran concentrations, which may reduce the molecular-weight dependence in certain dynamic parameters, the segmental interaction of dextran with water depends on the size of the dextran forming the transient network structure. In p.m.r.-echo studies, Tsitsishvili et aL5 arrived at a value of (M,),,, = 1300 g mol-l. In contrast to our findings they claim that (Ml),,, was essentially concentration independent in the polymer entanglement region.They also used a different method of calculating (M,),,, which relied on an obscure relation between the degree of hydration and the probability of centres of collision between dextran and water.W. D. COMPER, M-P. I. VAN DAMME AND B. N. PRESTON 3375 t o4 L e, h .. 3 1 o2 t I I 100 200 dextran concentration/kg m-3 FIG. 5.-Calculated values of (M,),ff from eqn (13) as a function of C, for dextran TI0 (-) and dextran FDR7783 (----). Values ofhy and 0: for a particular concentration have been obtained from the graph of their concentration dependence. RELATIONSHIP OF DYNAMIC DEXTRAN-DEXTRAN INTERACTIONS AND D E X T R A N-W A T E R I N T E R A C T I 0 N S T 0 T R AN S LA T I 0 N D I F F U S I 0 N COEFFICIENTS In order to establish the relationship of the various binary frictional interactions that can occur in the dextran+water system we also require an estimate of dextran-dextran interactions.This may be performed through a study of the intra- diffusion of dextran. The intradiffusion coefficient of dextran may be described in terms of frictional coefficients, when c, * 6 c,, such that14 Using the values off,? and DT we may calculate fi *, from eqn (14) (see table 1). The calculated values off, *, are found to be consistently negative over almost the whole dextran concentration range for both dextrans studied. The magnitude of this parameter is relatively lower for TI0 than it is for FDR7783. It is unlikely that experimental errors in the estimate of D, and D,t could account for the consistent behaviour of the value of f, *,.It could be argued in this study, however, that difficulties with theoretical analysis of the experimental results is due to the polydis- persity associated with the dextran fractions used. A serious error may then be introduced through an incorrect assignment of the value of M , in calculatingfly from eqn (5) (see also the Appendix). For example, if a & 50% error, at worst, was made in the assignment of M , (due to effects of polydispersity etc.) then calculated values of fly would also be expected to be ca. 50% in error at high dextran concentrations. Even so, the negative value off, *, would still be observed for FDR7783. In fact, due to the low degree of polydispersity for FDR7783 (with Mw/Mn = 1.32), incorrect assignments of M , inf,, calculations would be far less serious than estimated in this hypothetical case.For T10 dextran, introduction of f 50% error in M , will result in a positive or negative value off, *,. At this stage we emphasise that the values off, *,3376 DIFFUSION OF HTO I N DEXTRAN+WATER TABLE 1 .-EVALUATION OF THE FRICTIONAL COEFFICIENT f, *1 frictional coefficients/ 1 0l6 dyn s mol-l cm-' f1*1 v12 +fi *1> f 1 2 (subtract column 3 C/kg m-3 [from eqn (14)la [from eqn (5)] from column 2) 20 28 36 54 62 68 98 104 136 150 160 198 7 25 43 64 70 2.6 3 .O 3.2 3.7 4.1 4.2 6.1 5.3 6.1 8.9 7.5 17.7 dextran T10 3.0 3.2 3.5 4.2 4.6 4.9 6.4 6.7 8.6 9.5 10.2 13.3 dextran FDR7783 12 14 15 26 23 40 22 61 36 93 - 0.4 - 0.2 - 0.3 - 0.5 -0.5 - 0.7 -0.3 - 1.4 -2.5 - 0.6 - 2.7 + 4.4 -2 -11 - 17 - 39 - 57 a Experimental determination of Dr.are either negative or low in magnitude as compared with fly. This does realise the unexpected finding that in concentration regimes where solute-solute interactions occur in some form (as manifested by the molecular-weight-independent dynamic behaviour of transient statistical network structures in concentrated polymer ~olutions)~~ the concentration-dependent positive frictional factor resides primarily in solvent-solute interactions, as embodied in f12, rather than in dynamic solute-solute interactions. It appears that the latter type of interactions may dominate the thermodynamic term cli3pl/ac, in eqn (5). VALIDITY OF EQUATIONS DESCRIBING POLYMER MUTUAL DIFFUSION In practice, untested assumptions are frequently made to yield a relationship between D, and Dt.A degree of inconsistency and ambiguity has been associated with these expressions.18 A number of these assumptions can now be properly assessed as we have evaluated all three conventional diffusion coefficients in the dextran + water system. Eqn ( 5 ) may be expressed in the following f0n1-P which requires the assumption that f 1 2 f 2 1 = f 2 *2f1*1*W. D. COMPER, M-P. I. VAN DAMME AND B. N. PRESTON 3377 Eqn (15) may be reduced to a more familiar form as our studies on the magnitude of DZ and D l have established that for dextran concentrations up to 200 kg m-3 the condition D;cl < DZc2 holds so that As all frictional coefficients other than possibly f l have positive values, therefore, in assuming f 2 *2 to be constant, low magnitude or negative values offl *1 as found in table 1 will invalidate the assumption embodied in eqn (16) and therefore invalidate eqn (1 5) and (17).A sufficient but not necessary condition for eqn (1 5) and (1 7) to hold is the property of the solution which Bearman14 has called regular, that - D: - - 3 D; ul so that eqn (15) may be reduced to the form12 This equation has been used previously in the analysis of polymer diffu~ion.~~ 2o The results of dextran and water intradiffusion reported here together with measurements of the partial specific volume of dextran do not bear this relation out. The factor D t / D ; decreases with dextral1 concentration whereas fi2/fi1 remains constant.At this stage, it appears that expressions for mutual-diffusion coefficients of the form expressed in eqn (1 5 ) and (1 9) are not valid for dextrans and most probably not for other water-soluble polymers. APPENDIX EVALUATION OF THE THERMODYNAMIC TERM Clapl/acl I N EQN ( 5 ) Algebraic expression for the chemical potential of component 1 as a function of composition has been given by OgstonZ1 in the form of p,-& = RT(lnrn,+a2rnl+a,rn~+. . . ) (A 1) where rn, is the molality of component 1 (moles per gram of solvent) and a2, a 3 . . . are the coefficients expressing thermodynamic non-ideality. Conversion of the concentration units of eqn (A 1) to a molar scale, where rn, = c1/c2M2, and then differentiation with respect to c, gives the form of cli3pl/acl required such that A problem arises, however, in the determination of the coefficients in the virial expansion of eqn (A 2).It is customary to relate these to a standard virial coefficient form through the osmotic pressure equation. Since the osmotic pressure ll of a polymer solution can be written as then with the use of the Gibbs-Duhem equation and eqn (A 1) we have3378 DIFFUSION OF HTO IN DEXTRAN-kWATER or we may rewrite eqn (A 4) in terms of standard virial coefficients (which are related to the virial expansion of concentration terms in mass/volume units) such that IIc,8, = RT[c, + A,(Ml)2 cf + A3(Ml), c:+ . . .I. (A 5 ) We will now show that the relationship between the Ai in eqn (A 5 ) to the coefficients in eqn (A 2) may take different forms, which ultimately leads to different expressions for c, i3p,/'?cl.If the values of A,, A , etc. have been obtained by fitting the experimenta! osmotic pressure data to eqn (A 5) then by correspondence to eqn (A 4) we also state the relationships 2a3 and A - a2 A - - 2c, M , ( M , ) , - 3(C2M2), (M,? which on substitution into eqn (A 2) gives, with no assumptions, This equation has been used in the main text and e l ~ e w h e r e . ~ ? ~ ~ If, as is commonly assumed for dilute solutions, c2tj2 * 1 then in eqn (A 4) and (A 5) values of A , and A , take the form defined in eqn (A 6). Substitution of these values into eqn (A 2) will give c, ["I = RT[ 1 + 2A,(M,), c, + 3A3(M1)3 cg + . . . ] (A 8) acl T , p when c,~, N 1 is consistently Employed.Eqn (A 8) has been employed in various treatments of polymerl9 diffusion but is strictly only valid for dilute polymer solutions. This project was supported by the Australian Research Grants Committee (grant no. D68/16898, D2 73/14127 and DS 79/15252). We thank Gregory Checkley and Robert Kitchen for their expert assistance. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 T. C. Laurent, L-0. Sundelof, K-0. Wik and B. Warmegird, Eur. J . Biochem., 1976, 68, 95. R. G. Kitchen, B. N. Preston and J. D. Wells, J . Polym. Sci., Pofym. Symp., 1976, 55, 39. B. N. Preston, W. D. Comper, A. E. Hughes, I. Snook and W. van Megen, J . Chem. SOC., Faraday Trans. 1, 1982, 78, 1209. D. Eisenberg and W. Kauzmann, The Structure and Properties of Water (Oxford University Press, London, 1969). V. G. Tsitsishvili, V. Ya Grinberg, E. I. 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