Oct., 1936 MOLECULARSXZE DISTRIBUTION IN LINEARPOLYMERS 1S77 [CONTRIBUTION NO. 164 FROM THE EXPERIMENTALSTATION OF E. I. DU PONT DE NEMOURS& COMPANY] Molecular Size Distribution in Linear Condensation Polymers’ RY PAIJLJ. FLORY -H*O Introduction HO-R-COOH --+H(0-R-CO),OH All synthetic polymers and most natural ones and those formed from glycols and dibasic acids are recognized to be non-homogeneous substances - H20 consisting of mixtures of chemically similar mole- HO-R-OH + HOCO-R’-COOH ---+ cules which are distributed in size about an aver- H( O-R-O-- CO-R ’-CO).OH age, some of them being much smaller and some In general, polymerizations7 of the first type much larger than this average. By various meth- above may be formulated as ods, more or less approximate estimates of the (i) x A-B +A-B (A-B),-,A---B average molecular weights of such materials can where A and B represent functional groups and be made, but as yet information concerning the AB, or BA, the product of their reaction with one distribution of molecules amongst the various another. A simple molecule such as water may sizes is exceedingly meager. An adequate un- or may not be released in this process. If, as in derstanding of the fundamental nature of poly- the second example above, there are two kinds of meric substances requires comprehension of the reactants, A-A and B-B, three varieties of molecular size distributions as well as the em- molecules may be formed in the polymeric mix- pirical compositions and average molecular ture, viz., weights. (iia) If x, the total number of reactant mole- Methods have been developed for the experi- cules combined in the polymer molecule, is an mental determination of molecular size distribu- even integer tions in polymers by using the Svedberg ultracen- x/Z A-A f x/Z B-B + trifuge,* but they have so far been applied only to A-A (B-BA--A)(, --2)/2B-R a limited extent. Theoretical calculation of the (iib) If x is odd, either molecular size distribution is possible if the ki- (Xfl )A-A+(~’)B-B- netic mechanism of the polymerization process is sufficiently well understood. Such &lculations have been attempted for vinyl polymerizations by Chalmer~,~by Dostal and Mark,*and more re- (x+l) A-A + (’+)B-B + cently by Sch~lz.~ The present paper offers a theoretical analysis B-BA-A( B-BA-A)(, -s)/*B--B of the molecular size distribution in linear conden- Another case which might be considered is that in sation polymers formed by the inter-molecular which the functional groups are all alike, but this reaction of bifunctional compounds. The poly- may be taken as a special case of (i) in which A = esters prepared by Carothers and co-workers6 B. provide excellent examples of this type of polymer. It is characteristic of all of these polymeriza- They are of two kinds, those formed from hy- tions that they involve the coupling of two bi- droxy acids functional compounds to form a bifunctional com- (1) A preliminary account of part of this work was included in a pound of higher molecular weight, which in turn paper presented by Dr. Wallace H. Carothers before the Farsday capable of polymerizing with other molecules. Society Symposium, September, 1935 [Trans. FUYQ~QYSOC., 88, 39 is (1936)1. Here the reactions : monomer with monomer, mon- (2) (a) Svedberg, Chem. Rev., 14, 1 (1934). (b) Signer and Gross, omer with polymer, and polymer with polymer lielo. Chim. .Acta, 17, 726 (1934). and Signer, Kolldid Z., TO, 24 (1935), have calculated size distributions for polystyrenes from sedi- all involve the same chemical process. These con- mentation velocities in the ultracentrifuge. (c) Lansing and Krae- densation polymerizations are not to be confused mer [THIS JOURNAL, 67, 1369 (1938)I have derived methods for calculating a “non-uniformity coefficient” from determination of the with polymerizations proceeding by a chain mecha- sedimentation equilibrium distribution of a polymeric material in the ultracentrifuge. nism, for example, vinyl polymerizations,* or (3) Chalmers, ibid., 86, 912 (1934). (7) The reasons for classing bifunctional condensations such as (4) Dostal and Mark, Z. physik. Chem., B19, 299 (1938). these as true polymerizations have been set forth by Carothers.1 (5) Schulz, ibid., BSO, 379 (1935). (8) Semenoff, “Chemical Kinetics and Chain Reactions,” Oxford (6) Cnrothers, Chem. Rev., 8, 359-08 (1931). Univ. Press, Oxford, England, 1935, p. 444. 1878 PAULJ. FLORY VOl. 5s the polymerization of formaldehyde catalyzed by “x-mer” will denote a polymer composed of x seg- formic acid.9 ments, e. g., a hexamer is a polymer composed of The treatment which follows rests upon the six segments.12 fundamental assumption that the reactivity of a The following notation will be used: functional group, A or B, is independent of the WD = total number of segments = total number of A (or size of the molecule to which it is attached. If B) groups. N = number of unreacted A (or B) groups which remain the functional groups are separated by more than after the reaction has been in progress for a time, t. three atoms in the chain, this assumption is prob- NO- N = number of reacted A (or B) groups. ably not seriously in error; if only one or two (If desired, the N’s may be expressed in moles without al- atoms separate the functional groups in the mono- tering any of the equations which follow.) p, the extent of reaction, = fraction of the total number mer, then the reactivities of monomer and dimer of A (or B) groups which have reacted at time t may differ greatly although the difference be- p = (No - N),”o (1) tween dimer and trimer will be much less, between Fundamental Distribution Functions.-In or- trimer and tetramer still less, etc.1° If the re- der to derive the distribution functions, let us de- verse reaction, polymer degradation, occurs ap- termine the probability, II,, that a particular seg- preciably, it is necessary to append the assump- ment selected at random is part o€ an x-mer. tion of equal reactivity of all links, AB, with re- Picture a large group of partially polymerized spect to the reverse reaction, e. g., hydrolysis in segments laid end to end, thus the case of polyesters. It is also assumed that S the formation of cyclic compounds (as well as etc. -- - etc. b-a b-2 b-1 bl b) bs other side reactions) does not occur to an appre- the b’s designating the linkages, or potential link- ciable extent.” The errors arising from the as- ages, between the segments. Since all unreacted sumptions will in general be most significant in functional groups are assumed to be equally reac- the range of low molecular weights; for polymers tive, the probability that reaction has occurred to of high average molecular weight, where the dis- link the segments at any particular b is p and the tribution is broad and the low molecular weight probability that no linkage exists is 1-p. There portion is small, deviation from the calculated re- are x possible configurations, such as for example sults should not be large. that consisting of linkages at bl, bz. bx-l and un- Theoretical Treatment and Discussion reacted functional groups at b-, and b,, which Type (i) polymers and type (ii) polymers fulfill the condition that the segment S be part of formed from equivalents of A-A and B-B mole- an x-mer. Since each of these consists of x-1 cules can be subjected to the same treatment and linkages and two unreacted potential linkages, so will be considered simultaneously. The special the probability of the existence of each particu- and more complicated problem of the size distribu- lar configuration is px-’(l -p)z. The probability tion relations in type (ii) polymers formed from that any of the x configurations exists isl8 non-equivalents of A-A and B-B molecules 11, = qY-1 (1 - p)2 (2) will be relegated to a separate later section. (12) In condensations of type (i) a segment is identical with a Definitions.--A reactant molecule or its radi- “structural unit” as previously defined by Carothers [THISJOURNAL 61, 2548 (1929); ibid., 66, 5023 (1933)], but in type (ii) condensa- cal will be defined as a “segment” and the term tions a segment is one A-A or one B-B group whereas the struc- (9) Carruthers and Norrish, Trons. Fwvodoy SOC.,82, 195 (1936). tural unit is A-AB-B. For type (ii) polymers the above definition (10) Factors which cause change in reaction rate during the course of an x-mer does not agree with the previous definition according of the reaction do not invalidate this assumption if the rates of reac- to which an 2-mer contains x structural units. The above definitions tion of a monomer, a dimer, etc , are all affected equally. Thus, the have been adopted for the sake of simplifying the following analysis. increase in viscosity of the medium as the reaction progresses might (13) Kuhn, Be?., 68, 1503 (1930), has derived the same equation slow down the reaction, but if the rate of reaction of each molecular for the molecular size distribution in a degraded cellulose formed by species is slowed down proportionately the equality of the reactivi- the hydrolysis of infinitely large cellulose molecules, assuming that ties of all functional groups at any one time remains undisturbed. all linkages are equally hydrolyzable. That the two distributions (11) That the extent of ring formation in bifunctional esterifica- should be identical is obvious when one considers the equivalence of tions involving unit lengths beyond 7 is exceedingly small has been the formation of polymer molecules through random formation of shown experimentally in many papers published from this Labora- linkages and through random breaking of linkages already formed.