1566
TAKASHI FUJIIAND HENRYC. THOMAS
Vol. 62
SELF-DIFFUSION OF SODIUM ION IN AGAR GELS1 BY TAKASHI FUJIIAND HENRYC. THO MAS^ Contribution No. 1609 from the Sterling Chemistry Laboratory, Yale University, New Haven, Coqn. Received June $6, 1868
Self-diffusion coefficients of sodium ion in agar gels of four compositions, prepared with 0.05 N NaCl, have been measured at four temperatures from 2.8 to 35". With w the weight per cent. of agar in the gel, the results are reproduced in the range w = 0.4 to 3.0 by the expression D = (O.O156w-*/a- 0.0052) exp [ -(5930 - 1910w1/s)/RT]. The form of this expression for the self-diffusion coefficient is discussed.
Agar, the dried mucilaginous water extract of certstin seaweeds, is a familair enough material, but many of its properties are not a t all well understood. It is known to be a heteropolysaccharide, but its chemical structure remains in controversy and its molecular weight is unknown. Neither is it known whether agar is a linear or a branched polymer. The gels of agar with salt solutions, however, may certainly be regarded as infinite three-dimensional networks. The study of diffusion in gels as a means of elucidating their structure has a long history. References to earlier work are given by fried mar^,^ who studied the diffusion of non-electrolytes in agar gels and established in this case a linear dependence of the diffusion coefficients on gel concentration. .Early work on the diffusion of electrolytes has been improved by the use of radioactive tracers. Thus Salvinien, Marignan and Cordier4 have studied the diffusion of phosphate, using P32,in gelatin gels. Their results do not follow the limiting law of Onsager and Fuossb; neither does the extrapolation of the diffusion coefficients to zero gelatin content agree with the value for pure water. A similar effect is found6 with the limiting conductances of electrolytes in aqueous sucrose solutions. The coefficients of self-diffusion here reported show like behavior. Ionic diffusion in agar gels undoubtedly is influenced to some extent by the existence in the structure of fixed negative groups. According to Araki? agar is composed of the two saccharides, agarose and agaropectin, with a constitution similar to that produced in starch by amylose and amylopectin. Agaropectin, however, contains sulfuric and uronic acid residues, which contribute t o the gel a small cation exchange capacity. The influence of these groups in diffusion measurements has been emphasized by Freise.* The cation exchange properties of different agars have been systematically studied by C ~ r r i e who , ~ found the exchange capacity to vary directly with the sulfur content. ,
(1) The material for this paper has been largely taken from the thesis submitted by Takashi Fujii to the Graduate School of Yale University, in partial fulfillment of the requirements for the degree of Master of Science. The authors wish to express their indebtedness to the American Petroleum Institute which supported this work. (2) To whom any communications may be directed a t his present address: Venable Hall, University of North Carolina, Chapel Hill, North Carolina. (3) L. Friedman, J . A m . Chem. Soc., 62, 1311 (1930). (4) J. Salvinien, R. Marignan and S. Cordier, J . chin. phya., 51, 101 (1954). (5) L. Onsager and R. M. Fuoss, THrs JOURNAL,36, 2689 (1932). (6) J. M . Stokek and R. H. Stokes, ibid,, 60. 217 (1956). (7) Ch. Araki, Bull. Chem. SOC.Japan, 29, 543 (1956). (8) V. Freise, Z . physik. Chem. (Frankfurt) N . F . , 4, 129 (1955). (9) T. Currie, Chemistry and Industry, 116 (1955).
Experimental The work reported here was somewhat incidental to an investigation of self-diffusion in unconsolidated beds of minerals; no attempt was made to characterize the specimen of agar used. The materiallo used was especially prepared for microbiological culture media and was stated to contain "a minimum" of pigments and salts. A nominal exchange capacity was determined by treating 1.0-g. samples of the dry material with 25 ml. of Na(NaZa)Cl, 0.05 N , and shaking for eight hours. From the decrease in activity of the solution (from 15297 to 14069 counts per minute) the capacity of the dry material was estimated to be 0.11 f: 0.01 meq. Per g. The diffusion measurements were made in the manner recently described." The method depends on measuring the total radioactivity of a short rod (in this case, of agar gel) from which diffusion is taking place through a thin membrane into a bath at zero activity. If Q/QO is the fractional activity remaining at time t , then
-k
Ea
al/RDt { 1 - 6e-ar/ot -I- . . . ) + . . .
(1)
In this formula a is the length of the "short rod," i.e., the depth of a straight-sided cup containing the gel. The quantity 5 is the effective thickness of the retaining membrane; it is given by E = b(Dq/Drnqm) (2) in which b is the actual thickness of the membrane and D,q, D,, qm, are diffusion coefficients and capacities for gel and membrane, respectively. In practice 5 is treated as an empirical parameter, determined by the experimental conditions; its separate characterization is unnecessary. The use of the formula for Q / Q o is described below. The gels were prepared simply by adding slowly the required weight of agar to 10 d. of the warm salt solution, heating on a steam-bath for five to ten minutes, and cooling slowly. An apparently homogeneous gel, without air bubbles is obtained. The cup for the diffusion experiment was turned from Lucite to a depth of 0.564 cm. ( = (l/+i)cm.!); the inside diameter was 1.035 cm. Before the gelling of the agar was complete the viscous fluid was poured into the cup. After the gel had set, the excess was sliced from the top of t,he cup with a razor blade. A membrane of unwaterproofed cellophane (du Pont gauge No. 600) was adjusted t o cover the gel. The circle of cellophane is clamped firmly into place when the diffusion cell is assembled, as shown in Fig. 1 . The reservoir of inactive sodium chloride solution and the diffusion cell are first brought to temperature, a.nd the cell then clamped into place in the well of the scintillating crystal. The diffusion run starts with the flow of solution from the reservoir over the membrane covering the gel. The stream of solution, moved by a Sigmamotor Pump, supplies the infinite bath and also serves to maintain the temperature of the cell. The velocity of the flow was maintained a t about 1000 ml./min. During one run the flow velocity was increased by about 300 ml./min. for approximately a third of (10) Bacto-Agar. Difco Certified, Control 430418. (11) H. C. Thomas, Proc. N a t l . Acad. Sci., 4 2 , 909 (1956).
Dec., 1958
SELF-DIFFUSION OF SODIUM ION IN AGARGELS
the run and then restored to the lower rate. No change in the course of the &/&o us. G c u r v e was observed. It was concluded that the bath was adequately stirred. In any case, effects due to thin films of stagnant solution at the membrane are automatically cared for in the theory of the experiment. To a first approximation such a (steady) film would only change the value of E . The reservoir of solution was sufficiently large (3000 ml.) so that the remova.1 of all the radiosodium into the bath would increase its activity by an amount corresponding to only one per cent. of the background rate; in effect the bath was infinitely large.
Treatment of Experimental Results The experiment gives an apparently straight line iover a considerable relation between Q/Qo and d range after the initial stages of the experiment. This relation is, of course, only a n approximation and the slope of this line does not give in fact the true value of D. A convenient correction procedure may be developed as follows. I n formula (1) we retain two terms in each of the first two braces and the term E 2 / a d a . Differentiating and the resulting expression with respect to in placing this derivative equal to - ( 2 / a 4 3 which D' is the first approximation to D as determined from the slope of the experimental straight line, we obtain an equation which can be solved for D as the second approximation to the true value. The final result is
1567
3
-di dg,
diffusion cell: a, agar gel; b, cellophane membrane; c, Lucite cup.
Fig. 1.-The
W ' k
0.5
0.0 I
in which the arguments of the H h functions are as given in equation 1. The second term is the part of the correction due to the presence of the membrane; it may be positive or negative. The first term in effect corrects for the decrease in the rate of the diffusion as the process continues; it is always positive. I n applying these corrections, graphs were constructed of the various functions for convenient values of the different parameters. The corresponding corrections range from 16% a t values of D near 0.5 X to 3% near 1 X lo+. Results and Discussion The experimental results are summarized in Table I. When log D us 1/T plots are constructed for the data a t fixed agar content for 15, 25, 35", straight lines are obtained, according to the relation D = Doexp(-EIRT). Graphically determined values for Do and E are given in Table I. The lines for the various compositions all intersect a t temperatures between 5 and 11"; the diffusion coefficients are nearly independent of gel composition in this range of temperature. Experiments were made a t a temperature near 3". The results are not particularly reliable but do confirm that D depends but slightly on composition a t low temperatures. For fixed temperatures, when the measured values of D are plotted against the square root of the weight per cent. of agar in the gel the relation is found to be linear to within a reasonable estimate of the accuracy of the data. The points a t 0.4 weight per cent. are near the limit below which
1.5
1.0
I
2.0 I
I
0.02
. 0.01
6
0.00
0.5
1.0
1.5
2.0
w -I/a. Fig. 2.--0 = DOexp( - E / R T ) as function of gel composition: Do,0,left ordinate us. w-'/a; E, I (0.1 kcal.), right ordinate us. wl/a.
gel formation does not take place. Our method, in the form described, is not adapted to determinations in liquids, and we have 110 data in the region below 0.4%. The intercept of the line for 25", 1.25 X 10-6 cm.2/sec., is considerably lower than the value reported by Mills12for 0.05 N NaCl in water, namely about 1.31 X 10-6. There is, of course, no justification for extending the empirical square root relation below the gel point. I n fact, one might expect a stronger dependence on agar content in this region. Thus Wang's13theory for the diffusion of water in protein solutions predicts, in agreement with experiment, a nearly linear dependence, for the neutral molecule, on the first power of the (12) R. M. Mills, J . Am. Chem. Soc.. 77, 6116 (1955). (13) J. H. Wang, ibid., 76, 4755 (1954).
TAKASHI FTJJIIAND HENRYC. THOMAS
1568
Vol. 62
TABLE I COEFF~CIENTS OF SELF-DIFFUSION OF SODIUMION
IN
AQARGELS
D X 106, cm.*/aec. T;mP., C.
Wt. % Agar
1st
Approx.
35.0 f0 . 2 25.0 15.0 2.8 f0.2 35.0 25.0 15.0 2.8 f0.2 35.0 f 0 . 2 25.0 f 0 . 1 15.0 f 0 . 5 2.8 f0.2 25.0 f 0 . 2 15.0 f 0 . 3
5 5 5 5 3 3 3 3 1 1 1 1 0.4 0.4
Gels with 0.05 N NaCl 1.13 1.129 0.99 0.970 0.76 0.83, 0.54 0.592 1.21 1.231 1.04 1.056 0.82 0.860 0.55 0.621 1.33 1.400 1.12 1.131 0.84 0.882 0.56 0.617 1.15 1.174 0.89 0.917
25.0 f 0 . 2
5 3 1 0.4
Gels with 0.10 N NaCl 0.88 0.888 0.93 0.946 1.01 1.032 1.05 1.078
weight per cent. Evidently a study of the behavior of the diffusion coefficients for ions across the gel point would be of some interest. When we examine the diffusion coefficients, D = DOexp( -E / R T ) , as functions of the gel composition several points of interest become apparent. I n terms of the transition state theory of diffusion, DOmeasures the contribution to the diffusion GOefficient of X2 (kT/h) exp(AS*/R), in which X is the component in the direction of diffusion of the average elementary displacement required to produce a n exchange of two sodium ions; AS* is the entropy change for the formation of the sodium pair. If the entropy of activation is independent of the agar content, the presence of these obstructing molecules will manifest itself only by a decrease in the value of X, the longer elementary jumps being blocked. We would then expect X to depend inversely on the cube root of the agar content and hence DO to vary inversely as the two-thirds power of w. I n Fig. 2 it is seen that this expectation is borne out except for the most concentrated gel, However, the required change in X for the relatively small change in w amounts t o some 500%, an unreasonably large amount.14 We must, therefore, (14) We are indebted t o Professor P. to
US.
2nd Approx.
S. Lyons who made this point
DO
E, kcal./ mole
81.9
2620
233
3200
1050
4050
2330
4480
,
also suppose that the probability of the associated state also decreases in nearly the same manner with the agar content, perhaps to an extent corresponding to several units in the entropy. The activation energy is likewise smaller for the more concentrated gels, and as is seen in Fig. 2, E is an accurately linear function of w'/1. Thus E decreases just in proportion to the distances available for the elementary jumps. These observations enable us t o summarize all the data for 0.05 N NaC1, for w = 0.4 to w = 3, in the relation D = ( 0 . 0 1 5 6 ~ - ~-/ ~0.0052) exp [ -(5930
- 19lOw'/a)/RT]
(4)
Certainly the form of this equation depends on somewhat tenuous justification. It will be interesting to extend the work in the many obvious directions available. For example, on the basis of our experiments the relation cannot be correct a t temperatures below 11'; more detailed work is needed a t low temperatures. We wish t o express our indebtedness to Dr. Alan Sussman who was responsible for a very large part of the technical work in constructing the apparatus, which he did in the course of his own researches on diffusion in ion exchangers.
.'
