J . Phys. Chem. 1984, 88, 6469-6472
and if so, could I come to Leipzig during my summer vacation of 1929 to work with him on this problem. Debye replied that no one had developed a viscosity theory for pure liquids to say nothing of solutions, and, if I wanted to come to Leipzig, he would be glad to talk with me. So I went to Leipzig a t the beginning of July 1929. Debye was then running a Symposium on Polar Molecules and was too busy to spend much time with me, but I did show him our viscosity data for barium chloride solutions. H e was quite interested to see how well the data fitted eq 1. I also saw Onsager at the Symposium and told him about the square root term in the Jones-Dole equation. H e immediately said that he thought that a strong electrolyte theory could be worked out for it. Debye had to return to Zurich for a few days after the Polar Molecule Symposium and sent to me on July 7 a post card on which he wrote: “In the train I had time to think about your problem. Now, I know how it must be handled; and I have the impression that before the end of this month you will have the theory you want. I hope that in the meantime you will talk with Fakenhagen and study as thoroughly as possible the calculation of the ionic atmosphere, especially according to Onsager. This will be absolutely necessary if you are to understand what has to follow.” On returning to Leipzig, Debye asked me it I was familiar with the calculus of spherical harmonics. I was not so he turned the problem over to Hans Falkenhagen, one of his assistants. Falkenhagen began the mathematical development and reached a point where a solution to a set of four homogeneous differential equations had to be found. H e would suggest a solution to me and I would then test it out. After two weeks, we still had not found a solution so Falkenhagen took me to Debye’s fairly spacious office which contained a roll-top desk, a conference table, and a blackboard on the wall at the far end. Falkenhagen told Debye our problem. Debye reached into his desk for a card and immediately wrote on it the following:
(3)
6469
We then went back to the laboratory where my desk was and found that eq 3 was the correct solution to the set of homogeneous differential equations. It was really amazing that Debye gave us the answer to our problem in terms of a differential equation. This little episode illustrates what a brilliant man Debye was. After about another week of work on the theory, Falkenhagen came to me and said that he had finished the derivation, but the answer was null; that is, the interionic forces had no effect on the viscosity of the electrolyte solutions. He tried to commiserate me by saying “not every research works out”. My heart sank when he told me this because, if true, it meant that our whole trip to Europe (my wife and I went at our own expense) was wasted from a scientific standpoint. I said, “Have you told Debye this?”. Falkenhagen replied “No, we go down to see him now”. So we went to Debye’s office where Falkenhagen went to the blackboard to describe how he had carried out the final integration over the layers of solution during the viscous flow measurements. Falkenhagen had hardly begun when Debye jumped up, saying “No! No!” and pointed out Falkenhagen’s error. This was all in rapid German which I did not follow. At any rate, Falkenhagen repeated his derivation and then came up with an equation for the relative viscosity in which the constant A of eq 1 could be Galculated. The numerical result was shown later to agree very well with data for KCl s o l ~ t i o n s . ~ ~ ~ In the original paper published by Falkenhagen and me7 (for some reason unknown to me Debye’s name was not included as one of the authors), the theory was worked out for the simplest case; that is, for salts containing positive and negative ions of equal valence and equal mobilities. Later, Falkenhagen went to the University of Wisconsin and developed the theory8v9for solutions of ions of any valence or mobility. To conclude, it is apparent that Debye’s contributions were most important for the development of the theory of the relative viscosity of solutions of strong electrolytes. (5) Joy, W. E.; Wolfenden, J. H. Nature (London) 1930, 126, 994. (6) Joy, W. E.; Wolfenden, J. H. Proc. R. SOC. London, Ser. A 1931, 134, 413. (7) Falkenhagen, H.; Dole, M. Phys. Z . 1929, 30, 611. (8) Falkenhagen, H. Phys. Z . 1931, 32, 745. (9) Falkenhagen, H.; Vernon, E. L. Phil. Mag. 1932, 14, 537.
Liquid-Liquid Demixing inside a Rigid Network. Qualitative Features P. G. de Gennes Colkge de France, Physique de la Matiere CondensPe, 75231 Paris Cedex 05, France (Received: May 1 , 1984;
In Final Form: August 1, 1984) When a gel G is immersed in a solvent-solvent mixture AB, it acts as a random field on AB. We expect two main regimes for the G + AB system: (1) a reversible regime, characterized by a single (renormalized) correlation length K-l; (2) a glassy regime, where the A/B distribution inside the gel becomes dependent upon the sample history. The glass transition point T . is reached when RLincreases up to a certain characteristic length I, dependent on the gel parameters (mesh size L, selective adsorption A). Two major properties of the glassy state are (a) the threshold injection pressure II, required to force a B-rich phase inside an A-rich gel and (b) the spreading coefficient D, giving the macroscopic width (2D,t)’I2 of the AB boundary region after a long time t of contact. We present some conjectures on the scaling structure of II, and D,.
I. Aims Certain, partly compatible, fluid mixtures AB (e.g., lutidine/water, or isobutyric acid/water) can saturate a gel G (agarose, polyacrylamide). Systematic measurements on G AB have been carried out recently by a joint action between Pittsburgh and Zurich.’
+
(1) Goldburg, W.; Maher, J.; Pohl, publication.
D.; Lanz, M.,
to be submitted for
0022-3654/84/2088-6469$01.50/0
These systems are often highly complex, but they deserve attention. On the experimental side, they offer new methods for probing gel structures, inhomogeneities, etc. On the theoretical side, they provide one example of randomfields: the randomness (2) Recent reviews on random fields: Aharony, A. “Proceedings of Magnetism Conference, Kyoto, 1982”. Imry, Y . “Proceedings of 15th IUPAP Conference (Statistical Mechanics), Edinburgh, 1983”. For relations with gels and porous media see: Brochard, F.; de Gennes, P. G. J . Phys., Lett. (Orsay, Fr.) 1983, 44, 785.
0 1984 American Chemical Society
6470 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
de Gennes and think of the system as inscribed on a lattice (lattice parameter a). The free energy density, in the Widom approximation: reduces
to
U is comparable to KT, (Tco= critical point of the gel-free AB system). 5 is the correlation length for the gel-free AB system. Figure 1. Selective adsorption of A (from an AB liquid mixture) on a rigid gel.
is built in during the fabrication of the gel. Networks with strictly periodic structures should have very different features when immersed in an AB m i x t ~ r e . ~ Up to now, most of the attention has been focused on the high-temperature behavior, where A and B are miscible (for definiteness we use the language appropriate for a “normal” AB consolute point: low temperature favor segregation), and where the measured properties (mainly light scattering) appear more or less independent of the sample hist0ry.l A naive (decoupling) description of this regirne1s4is rederived, for tutorial purposes, in section 11. The most novel features, however, occur at lower temperatures. Upon cooling down, the light scattering pattern becomes frozen.’ On the theoretical side, a stimulating discussion of the freezing process has been given recently by Villah5 In section I11 we present some conjectures on this glass transition. We also discriminate between two regimes in the glassy state: (a) If we go deep into the glass phase, the bare AB interface has a thickness ([) which becomes smaller than the mesh size (L) of the gel; in this limit we can use a certain analogy with biphasic flows in porous media6 and define some important hydrodynamic observables (section IV). (b) The opposite case (6 > L ) is what we call a glassy continuum and is briefly presented in section V. All our discussion is restricted to rigid gels (undeformable networks). In the opposite limit (flexible chains) many further complications are expected.’ In practice, the agarose fibers are probably quite rigid. Polyacrylamide networks may also be rigid (in low salt) because of partial hydrolysis, leading to ionized sites and Coulomb repulsions. The gel G has two main local effects on the AB mixture: (a) One effect is preferential adsorption of one component (say, A). A long series of studies on preferential adsorption has been carried out by Benoit and Dondos.8 The role of preferential adsorption is illustrated in Figure 1. (b) The other effect is reduction of the AB couplings in the vicinity of the G monomers. Among other things, effect b causes a shift of the consolute temperature T,. In the present paper we omit this shift and we assume that effect a is dominant, as indeed suggested by many systems studied by Benoit and Dondos. This allows us to describe the cloud of A near G in terms of a simplified free energy-what we usually call the “Widom approximation”, to be described in the next section. 11. Reversible Behavior
We shall describe the AB system in terms of its magnetic analogue, with a magnetization
(CA,CB = number densities-of A and B)
5 = ~ [ T , , / ( T- ~ ~ ~ ) 1 ~ / ~ (For T C T, the W/t2term is changed in sign.) Note the absence of any M4 term in eq 11.1: this leads to the desired form of the coexistence curve for the gel-free system Mo(T)
where A is a dimensionless measure of preferential adsorption, and C, is the local gel concentration. For more complicated gels (e.g., agarose) where the network is not made of single chains, but rather of many-chain fibers, a slightly more general formulation would be required (Le., replace CG by the concentration of monomers which are actually exposed to the solvents) but we shall not insist here on these minor complications. It is convenient to split h(r) into its space average ( h ) plus a fluctuation term
h(r) = ( h ) + hl(r) H + ( h ) = HI
(11.3)
The major effect of hl is to induce a “magnetization cloud” around each network portion. If one takes H I = 0 and T C T,, this cloud is described by (11.4) ( R = Ir
- r’l)
where K-l is a renormalized correlation length (different from 4 ) which we shall determine self-consistenty. To do this, we decouple the M6 term in eq 11.1, writing M6 (constant)(M2)Zil@. We then see in eq 11.1 that
-
K2 = 52 + (W)2a-2
(11.5)
In eq 11.5, and in all that follows, we drop out symmetrically all numerical factors. We compute (@) through eq 11.4 and focus our attention on weak preferential adsorption A T,,, K-' 5. But close to T,, K-' 1. It is formally possible to extend eq 11.10 below T , (this essentially replaces Ezby -E2). However, as we shall see, a glass transition occurs, and this low-temperature extension, based on assumptions of thermodynamic equilibrium, is probably not meaningful. Equivalents to eq 11.10 have been mentioned long ago.2 The basic decoupling scheme is very rough, but probably sufficient for our purposes. Clearly the interesting region is KI 1. Thus, our requirement K L E(T) and that the analogy between gel and porous medium is correct. What do we expect in this case? (1) If an A-rich gel sample ( M r Mo) is exposed to a B-rich ( M = -Mo) fluid, from the arguments summarized in ref 6 , we expect no macroscopic entry of B, until the capillary pressure difference n = PB - P , (IV.1)
-
-
reaches a certain threshold II = nc(in our model the difference n is related to the "external field" H by 2M0H = II). The order of magnitude of II, is given by the flip process of ref 12:
n, Thus
= MohaL-*
-
€IckTcA/(Lza)
(IV.2) (IV.3)
(2) When n > II,, we expect a macroscopic entry of B. This, however, takes place slowly in time, in the form of a penetration (1 1) Belanger, D.; King, A.; Jaccarino, V. Phys. Rev. Lett. 1982,48, 1050. (12) Brochard, F.; de Gennes, P. G. J . Phys. (Lett.) 1983, 44L,7 8 5 .
J . Phys. Chem. 1984, 88, 6472-6479
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‘n
V. Glassy Continuum Let us now reverse the inequality which was assumed in section IV, and take L < ,$(T), by choosing either a tight gel (small L) or a temperature T in the lower vicinity of T , (t large). More
(1)-
precisely we shall require
A rich liquid x&+-
Figure 3. Macroscopic entry of an A-rich liquid into a B-rich gel, at temperatures where the G + AB system is glassy. The scaling law for the front width A( T ) is given in eq IV.4. front, qualitatively shown in Figure 3. The shape of the front depends on two distinct percolation processes and is rather complexS6 But there is one simple feature: the macroscopic width of the front is an increasing function of the time t of contact; in a purely passive situation (no sweeping, no gravitational effects) the width A ( t ) should be proportional to the square root of t : 6 A2(t) = 2D,t
(IV.4)
This defines a “macroscopic diffusion coefficient” 0,. The scaling structure of D .,.. is Dm
-
IIcL2/7s
-
kTcA/(a~s)
(IV.5)
The two quantities II, and D, are the key parameters for irreversible entry in the gel. Their experimental determination is nontrivial (for instance, the threshold II, is expected to be sharp only for large samples and long measuring times), but feasible. In principle, from II, and 0, one could extract both L and A.
L
