DILUTEGELLINGSYSTEMS
August, 1958 a = 22.5 A. b = 1.35 u = 3.436 x 10-8 (ohm cxn.)-l D = 79.26 q = 0.9370 centipoise p = 3.604 X cm.2/volt sec. 1 ; ~= 84.6 millivolts 1; = 48.2 millivolts z = 7.4 x 10-4
The results of these calculations are given in Table I. The electroviscous effect predicted by Smoluchowski, Krasny-Ergen and Finkelstein and Cursin is more than one order of magnitude greater than the observed values, even in 0.2 M sodium chloride. Booth's theory agrees with the observed values within experimental error. Mobility data obtained at 25" by Stigter and Mysels for this system4 give essentially the same results when corrected to 23". Similar calculations were made for dodecyltrimethylammonium chloride using data obtained in this Laboratory at 23" for the mobility, micellar weight and critical micelle concentration.6~~This detergent was chosen because it has micellar weights about 4oy0 lower and zeta potentials about 25Y0 lower than those of sodium dodecylsulfate at the same ionic strength. The theoretical and experimental valuesa for the electroviscous effect are listed in Table I. Again the agreement between Booth's theory and the observed values is within experimental error. Hoyer and Greenfield' recently have measured the mobility, of dodecylamine hydrochloride micelles a t 25" in aqueous solutions of 0 to 0.048 M sodium chloride. At this temperature, the detergent has a very limited solubility in water and may even have been precipitating during some of the measurements in 0.048 M salt. The mobilities were corrected to 30" by means of the relation
969
between mobility and temperature determined between 18 and 35" by Hoyer and Greenfield for dodecylamine hydrochloride in water. Zeta potentials were calculated using data obtained for micellar weights at 30°,6 a temperature at which the detergent is much more soluble. The micellar weights are about 100% higher and the zeta potentials about 20% higher than those of sodium dodecylsulfate at the same ionic strength. In Table I are listed the theoretical electroviscous effects and the experimental effects at corresponding salt concentrations estimated from the literature values by interpolation.6 The agreement between Booth's theory and the observed values is good considering the approximations made in the calculation and the experimental difficulties at the higher salt concentrations. A comparison between the measured electroviscous effects and those predicted by Booth for each of the detergents as a function of the concentration of electrolyte at the critical micelle concentration is shown in Fig. 1. Thus, by means of Booth's theory for the electroviscous effect, data concerning the micellar weight, electrophoretic mobility and intrinsic viscosity for three ionic detergents in aqueous solutions of sodium chloride can be shown to be self consistent within experimental error. The assumption made by Booth that terms in the expression for the electroviscous effect of higher order than the second power of the zeta potential may be neglected appears to be adequate even though many of the zeta potentials were about 100 mv. The theories of Smoluchowski, Krasny-Ergen and Finkelstein and Cursiii appear to be inadequate for these systems; they predict electroviscous effects more than one order of magnitude greater than those observed.
DILUTE GELLING SYSTE,MS. I. THE EFFECT OF RING FORMATION ON GELATION BY R. W. KILB General Electric Research Laboratory, Schenectady, N . Y . Received April $8, 1968
Flory's method for the calculation of extent of reaction a t gelation of polymer systems is extended in an ap roximate manner to include ring formation. I n some cases the formulas are uite simple. The effect of rings is to m a t e the gel point dependent upon the dilution of the system so that at higher dJutions gelation occurs at higher extents of reaction. It is found that short and flexible monomer units give a greater volume dependence than long and stiff monomers.
The problem of ring formation in polymers has been studied previously by Jacobson and Stockmayerl and by Harris.2 However, the scope of situations treated by them is somewhat restricted. The former authors' studied polymerization of difunctional monomers, and Harris2 considered the case of pure f-functional monomer. The problem of particular interest to us is the dependence upon volume of the extent of reaction at gelation for polyesters or similar systems. These (1) H. Jacobson and W. H. Stockmayer, 1. Chem. Phys., 18, 1600
(1950). (2) F. E. Harris, ibid., 28, 1518 (1955).
usually involve two different (Type X-X and Y-Y) difunctional monomers as well as an f-functional monomer (Type f x ,f 3 3). Stockmayer3 has given an excellent treatment of this system assuming no ring formation, thereby eliminating a volume dependence. An extension of his method to allow for ring formation soon leads t o formidable mathematical diffi~ulties.~ Flory's methods of calculating the extent of re(3) W. H. Stockmayer, ibid., 11, 45 (1943). (4) F. E. Harris, private communication. ( 5 ) P. J. Flory, "Principles of Polymer Chemistry," Cornell Univ. Press, Ithaca, N. Y., 1953. p. 360 #.
R. W. KILB
970
action a t gelation is more amenable to an approximate treatment which takes ring formation into account. Essentially, Flory calculates the probability of adding another f-functional monomer to a polymer. The polymer gels when this probability is unity. Our correction consists of a factor allowing for the possibility of reaction of the growing chain with an f-functional unit already in the polymer, and the concomitant introduction of a ring. The results of Stockmayer and Jacobson' may be used to obtain an approximation for the probability that a chain of n-units will form a ring. For the present purpose of counting, we consider a Type X plus Type Y monomer as a single unit (denote this by XU). If C, represents the number of chains of n units, and R, the number of rings, then1 Cn = Axn
(la)
R, = BVxnn-s/2 (1b) where A and .z: are parameters determined by the reaction conditions. We see from equation 5b of ref. 1 that A is nearly proportional to the number of moles, m, of YY units present. Thus we may replace (la) by Cn A'mxn (la') also'
-
B = (3/2~)'/#2ba
(2)
where v is the number of chain atoms per XY unit and b is the effective link length. We therefore take as the probability for forming an n-unit ring rn =
R, ~
Cn
+ Rn
N
Cn A'm = KDn-'/z
(3b)
up
where v, is the volume of polymer formed, and D = V/v, is the dilution of the system. This assumes that the rings are constantly forming and breaking up. If this is not the case, then the factor n in equation VI11 of ref. 1 must be omitted, so yielding R, = BVx"n-a/a (IC) Proceeding as Flory did, we now calculate the probability of reaching from one to another ffunctional unit. Let p represent the ratio of X's (reacted and unreacted) belonging to branch units to the total number of X's. We pick a chain end on the polymer and move inward until we reach an f-functional unit. The segment of the chain from the first to the second f-functional unit may be represented by (for f = 3) X
X
>X
Y-Y
[X-x
Y-YIix-
1/3. If no X-X monomers are present, p = 1 and equation 10 reduces to
K~/KZ
=
(mzvpl/mlvpz)( A ~ ’ / A ~ ’ ) v ~ 8 / z b z a / v l a /(15) ~bla
From reference 1,it appears that A2’/Al‘ is genermay be ally of order unity. The ratio m2vpl/mlvp~ determined from the gel system. Thus in comparb ing two systems which have the same effective link-lengths, bl = bz, but one has more chain atoms per XY unit, 1 2 > v], we find ~2 < KI. Therefore a smaller volume effect is obtained for longer XY units. Also, if in the two systems the number of chain atoms per XY unit is the same, v2 = vl, but in one case the effective link-length is greater, b.>bl, then Generally speaking, addition of side again K Z < K l . groups t o a polymer chain leads to stiffer chains and therefore greater link-lengths b. Thus one predicts that addition of side groups to the chain leads to smaller volume effects. The above remarks on the ratio K J K Z are necessarily qualitative as the ratio Al’/Az’ is somewhat, indeterminate. Nevertheless, the data of Price, Gibbs and Zimm7 and of Prices bear out the above description reasonably well. If the system consists of Type X-X and f X units capable of reacting with one another, the above equations apply after p y has been set equal to unity. The author wishes to acknowledge the advice and criticisms of Drs. F. P. Price and B. H. Zimm. (7) F. P . Price, J. H. Gibbs and B. H. Zimm, THrs JOURNAL,62, 972 (1958). ( 8 ) F. P. Price, ibid., 68, 977 (1958).
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