Copolymerization

A critical discussion of the mechanism of formation of copolymers by addition polymeri zation is presented. It deals mainly with the following three fundamental aspects: lCirst, the quantitative treatment of the reaction starting with a scheme consisting of initiation , growth, and termination mechanisms. Expressions for the instantaneous and total copolymer composition and for the over-all rate of reaction as function of monomer composition and of cOlwersion are derived in terms of quantities characteristic of the rpaction. Graphical and numerical methods for the determination of these parameters from experimental data are gi\'en in detail. The statistical distribution of molecular weights and compositions in the product is considcred in rc>lation to the constants of the rcaction and to thc analogous ca~e of simple polymers. Second, methods for the anal.\'sis of copolymer compositions are discllssed and experi mental l'Csults are summarized. Reactivity ratios describing the bella\'ior ill gro\\,th of a given radical toward a pair of monomC'I'8 are tabulated for a series of systems. Third, these results are interpreted on the basi~ of resonance and of electrostatic and steric cfTects as encountered in the study of certain organic reactions. In addition, degradation of cOP01.\'111erS is brieny considC'recl in the ligh t of the possible types of sequences in the chaill. A numerical relation bctween yicld and copolymer com position is derived. The prolJlc>m s rcmaining are principall.v the fo]Jo\\'ing: of copol~'Jllcr analy~i~, determination of o\'er-all rate::; of reaction and of individual rate con stants, and a more Fundamental correlation. between structure of monomers and behavior in copolymerization. Also, systematic data on the thermodynamic and rate propC'rties of copolymer solutions should be of great interest, and studies of th~ bulk properties and their rdation to copolymer structure represent a field where research has onl,' recently been initiated .


Introduction
Following the extensive experimental and theoretical attack on the general problem of chain polymerization reactions, r ecent years have brought a series of fundamental investigations regarding copolymcrization reactions. A framework: for the analysis of over-all monomer consumption and resulting change in average polymer composition has been created [1,18,26,31]2 and tested cxperimcntally [5,18]. Equations for the size and composition distributions have also been developed [26,28]. However, no systematic experimental results on such distributions are avail-able at present. Finally,intcrprctation s of the differcnces in relative reactivities h ave been made on the basis of the electronic structul'C of t he individual monom ers [L5, 24J. It is thc purpose of this artiele to r eview thc main points of these t heoretical and experimcntal investigations. In respect to the former , strcss will be laid on those aspects which also have been examined expel imentally.

General Remarks on Chain Polymerization Reactions
It is well established today that initiation , growth, and termination arc the principal, although not n ecessarily the only mechanisms that determine the kinetics of chain polymerization reactions . The rates of these individual steps vary widely. The growth reaction is the fastest. The initiation, which produces, by one means or another, out of a stable monomer an activated radical is by far the slowest step, whenever long chains arc form ed. Otherwise, the supply of active monomer wou ld b e too large compared with the demand of the growth reaction for stable monOlner .
The crucial step then, to begin with, is the production of a certain number of radicals able to grow before they are terminated. Their to taJ concentration is determined by the initiation and termination only, since the growth merely changes their molecular weight. If they arc terminated much faster than they are produced, an equilibrium is established. The exact condition for this to b e true r equires the mean life time of the active radicals to b e small in comparison with that of stable monomer. This defines a quasistationary state and allows the expression of the "steady state" concentration of fre e radicals by means of an "equilibrium" constant given by the ratio Equilibrium _ rate of production of fre e radicals.
constant -rate of destruction of fr ee radicals This additional, and in most cases of interest, valid assumption simplifies considerably the quantitative treatment. It is then possible to develop completely the kinetics of the polymerization reaction and the r esulting molecular size distribution [12] on the basis of a postulated react ion m echan ism. It will be shown here how to carry this program through when two or more competing monomer species are present. Additional problems then arise. One of the most important questions is concerned with the change in average polymer composition with changing composition of monomer residue . The mean composition depends on the relative rate with which the different species enter the growing cha in and hen ce, upon the relative growth rates, if we exclude t he in significa, nt numb er of (Iimers, kimel's, and otiler very short cha in s. In ad dition to t he inhomogeneity in r espect to clHl,in length necessarily occllring in polymerizing systems, t h ere will now exis t also ilu c tu a tions of th e com posi tion from ch ain to cl19 in . These flu ctuations depend upon t hose in the long cha in radicals. All three steps are essential for the determination of the d istribut ion curves.

. Basic Reaction Scheme and Equations
After these general remarks, we turn to the treatment of t he copolymerization of a binary system, a8suming t he simplest possible rea ction schem e, as indicR ted above. No kinetic stud ies exist at presen t th at would necessitate th e consideration of fl dd i tionfll elem en tary acts, as is the case for one-component system s. In v iew of what was sa id previou sly thi s would not efl'ect the calcula t ion of aver age polym er composit ions. In what follo ws, let n TS (A) be the numb er of growin g rad ical cl19ins, cadi of whi ch contflins altogetllCr r units of component A flnd s units of componen t B, while hav in g flJl act ivfltecl end consisting of an A-type monom er . nT. U;) is then corresponci in gly d efinecl . N T"' represents t he 111lmbel' of stabil ized ch a in s of specifi ed co mposi t ion . o is a catalyst moleCl ile and R fl rfld icfl l produced by its decomposit ion. Th e fo ll owi ng sch eme m ay then be cons idered . In Wl'l t ll1g L h e initiation equation we have considered two types of activation. In th e ca talyzed activat ion tb e el em entary act consists in a decom po i t ion of th e ca talyst C. In th e equations for growth and term ination, we h ave differentiated bet veen A -A , A -B, B-A, and B -B addition of mon omer to r adical and radical to radical, resp ect ively. The rates of co n umption of monomer arc given according to eq 1 b y Th e te rms I A and I s are omitted for a catalyzed.
An analogoll s equation results for ~nTs (R) . Equation 3 exp resses the Ifl ct tlmt A-type ]'sd ica ls are produced by addi t ion o r A-type monom er to B-type racli cnls, and t hey are destr oyed by ncld ing B-type monomer and by termination wi th an Aor R-type radicsl. The terms contributed by the chain-breaking res ction are sm all comp ared with the grow th term s in eq 3, if long cha ins are to be form ed. H ence, th ey m ay be neglected , ancl we obtain the simple relation : It expresses the fact that free radicals wi th an active end A are produced ss rapidly by nddi tion of monomer A to free radicals with an active end B as they are destroyed by addition of monomer B to free rad icals with an active end A.
3 . C omp ositional Relationships: Average Composition, Relation to Conversion Insertion of eq 3a into eq 3 leads to th e following relation for the change in composition of monomer residue:

dA A k gA( B ) k gA(A)A+ lcgB(A) B dB = B lc-;B -(A) k gA(B )A + lcgB( B ) B' (4)
It has been assumed in eq 1 that the rate constants are essentially independent of chain length, an assumption commonly made in chain polymerization reactions and shown to be true in polycondensation reactions [llV Furthermore, they are independent of chain comp.osition T,S in eq 1. Herington and Robertson [12] have established equations that allow in principle a deduction of such a dependence from molecular-weight distributions. Analogous relations for copolymerizing systems have been developed by Simha and Branson [26]. However, the equations are too complex and experimental results nonexistent to merit further discussion here. It may be noted only that eq 4 is unaffected by any such assumption. If the growth rates depend upon the composition, then the constants kpL (1I1) in eq 4 represent mean values averaged over the radicals nrs(M) with the particular end M , that is, for instance: gives the relative rates of consumption of the two monomer species A and BundeI' the approximations stated previously. It also represents the instantaneous average composition of copolymer formed at an instant in which the monomer residu e consisted of A moles of species one and B moles of species two. This composition can not depend on the absolute magnitude of the growth rates kg but only upon the relative rates of addition of each monomer species. Hence, the following parameters are defined [18:] The symbols in the brackets indicate the modes of addition to which the constants () and p, refer. 4 1 H. W. Melv ille (lecture presented at the Kational Burean of Standards on April II, 194i) find s that growth and termination rates decrease but slightly with increasing chain length in t he polymerization of vinyl acetate in the liquid phase.
Both parameters express the growth rate by means of addition of monomer of the same kind to an activated end relative to the addition of a monomer of the other kind. D efining furthermore the mole ratio A lB as z, eq 4 is transformed into ) or dA+dB= l + z (lTz + l) .  This equ ation forms at presenL Lhe basis for th e determination of th e parameLers (J and J-L from an analysis of th e copolymer composition . Applications will be discussed in Lh e next sec t ion . First, h owever, we shall consider som e general properties of Lh e function r epresented by eq 6. Figure 4 sh ows a series of plots of log BolB vers us log zlzo     The final polymer form ed consists, in these examples, en tirely of species A, Z= (X) , for B = O.
When ,u> 1 and <T > 1, A -A and B-B linkages ar e more probable than cross-overs. Th e corresponding curves exh ibit an upward curvature and possess a v ertical asymptote shifting to th e left as <T increases . In other words, the final polymer form ed is a copolymer of fixed composition. Und er such conditions the ra tios between the r ates balance the con cen tration ratios in the monomer r esidue in such a way that the polymer composition equals the composition of the monomer mixture or R elation 4a then gives for the cri tical composItion Zc Thus <T and ,u must both b e eith er smaller or larger than unity . The fac t th at ther e exists a mix ture of definite composition that copolymerizes wi thou t changing its composition, suggests an analogy with the familiar azeotropic mixtures often en counter ed in distillation processes [31] . One can construct. curves analogous to distillation curves by considering th e sum of th e rates of consumption of

Journal of Research
The curves for t h e m onomer (lIf) represent FA + VB as a function of A /A + B =z/z+ l , according to th e expression above. The abcissas of conjugate po ints on the two li ncs P an d 1' vl indicatr th e composit ion of th e copolym er and th e I composit ion of th e co rresponding m onom er r esidu e, resp ectiv ely. F ig ure 6 d ep icts an "azeotrop e.)) It will bc noted th at h er e th e composit ion X does not correspond to an extremum in th e curvcs, which represents a n ecessary th ermodynam ic con d ition for azeotropic boiling mixtures.
Clearly th e second intersection in figure 6 docs n ot indicate a common composition for polymer and mon om er. The di sc ussion of th ese curves follows I oth erwise familiar lines. If !l and (T are both I smaller th an unity, th at is, if cross-co mbinations to.-are preferred, one derives from eq 6 th at z n ever ? reach es th e crit ical value Zc b u t app ro ach es zero I or infini ty, depcnd ing on wh eth er Zo is r esp ecLively smaller or greater th an Zc . If boLh react ivity ratios exceed un ity, th en th e azeotropic com position is grad ually approach ed.
It is of inLcr est to follow th e cha nges in compositi on takin g place in th e course of th e copolymerization . F igure 7 depicts Lh e relation be l wecn th e instan Laneous and total copolymer composition and th e per centage conversion. S uch curves are computed in th e following manne r. Copolymerization th e weigh t of polym er at a given instan t and th e in it ia l we igh t of t be monom er m ixture is given by Lh e equation : J3/Bo a nd z/zo h ave th e sam e m ea nin g as in eq 6. 11 1 a nd 1\12 a re th e m olee ula r weigh Ls of t he two m onomers. W ith th e a id of eq 6, B /Bo can be eliminated an d wp/wo obtai ned as fun ct ion of z/zo. The instan taneo us copolymer com posi t ion is directly g iven by eq 4a. T Ile total composition resulting up to a give n instant equals : a nd is again obLain ed w ith t be aid of eq 6. As is obvio us, t he differ ence between insta n taneo ll s and total values inc reases w it h tim e a nd is mo re p ronoun ced fo r systcm s with wi dely chA'el'en t react iv ity ratios . In ease an azeotr ope exists, and th e in it ial com posit ion h as been s uila bly cb osen, tlle ch a nges are sligh t as sh own i 11 curves 0 and Cf.
On th e wh ole, t he LoLal copolymer composition cloes not cha nge very m uch over a wiele range of co n vers ion. Th ese vari at ions can b e eliminated by compensatory addi t ion of th e more active sp CCLes during th e co urse of th e polymerization. 4. O ver-all r ate of reaction I t is eviden t th at t he calcul ation o f t be aver age copolym er composit ion d ocs n ot req uire a kn owledge of th e total concen tr ation of I n 'e r aeli cals b ut mer ely th e ratio of Aand B-type rad icals. Fo r th e over-all r eaction r ate, h owever , th is infor maLion is required and ob tained in th e follow ing m anner. In a steady state and for t be s implest case of a catalyzed reaction, we can wri te

t(A* + B* ) = I G-lc t(A,A)A *2_
wh ere Com b in ation of eq 3a, 3b , and 2 th en leacls to th e following expression for the over-all rate of copolymerization [26]:

a -1e ,(A,A ) le gB (A ),(3 ---;; Ic gB (B ) !c t(A,A )'
It should b e noted again that in this derivation the initiation is d escribed by a single constant I. This restricts the generality of this equation. For it is possible that the rate of production of the initial radicals varies vvith the composition of the mixture. In the most gen eral case of the r eaction scheme described, nine cons tan ts altogether would be needed to describe the process completely.

528
No use has as yet been made of th ese relations. Rate studies on th e system styrene-methyl methacrylate h ave been presented by Non'ish and Brookman [21]. However, they have been interpreted on th e basis of an equation that assumes the concent.rations A * and B* to be independent of the monomer ratio z . P lots of the rate (eq 2a) as function of the monomer composition A are shown in figure 8 for a few extreme cases.

. C omposition, Size, and Intramolecular Seq uence Distributions
For the considerations hitherto presented, it is necessary only to know the total concentrations of radicals ~nrs (A ) and ~n rs (B ) , regardless of size and composition. In order to obtain the distribution of polymer sizes and compositions, we must consider the mode of production of individual ~ radical chains of specified chain length and composition. From the postulated m echanism (eq 1), we find for the rate of production of these radicals: Equation 3e may be compared with the corresponding eq 3 for the total concentrations of Aand B-radicals and interpreted in the same manner. The positive terms refer to the growth of smaller chains to the desired size by monomer addition, and the n egative ones measure the rate of disappearance of the radicals in question by either further growth or termination . Equation 3c holds for rand s both larger than unity . The rate of production of species nlO (A ) and nOI (B ) is governed by the rate of initiation such that in the first set of eq 3c the positive t erms are replaced by I AA and in the second one by IBB or corresponding expressions involving the catalyst concentration. In a steady state the left-hand sides of eq 3c vanish . The foll owing q uan tities may be defined:

Ie teA, A ) n rs( A )~n ik( A )-
It will be noted that w(A) represents t he probability of formation of an A-A linkage by propagation, w(B) that of a B-B bond. x measures the probability of oeeurrenee of B -A and A -B linkages relative to that of A -A and B -B bonds. The solu tion of eq 3c has been shown by Simha and ,Branson [26] to be The exponential factors indicate the probabili ty of occurrence of these configurations. The summation is then carried out over all possible internal I arrangements compat ible with the condition of having r A-un its and S B-units. It is taken between th e extremes of having one long sequence of A -uni ts followed by one sequence of B-units, (j = 1), and the opposite extreme of a checker board arrangement of these two monomers. Th e I n terms then sLand for chains in itiated by a Bradical. The meanin g of th e last eq uation may be seen a lso by speciali zing to th e case U!l = 1, in which propagation is independent of the nature of the growing end . Equation 3d th en reduces to [10] Copolymerization n rs= n Ts( A ) + n Ts( B) = [ k g:CA) C+ ~-1)+ Clearly Lhe first term en umerates all ways of obta inin g Lhe polymer n TS from a nucleus nlO; th e seco nd L(' rm from a nucleu n Ol o From t he J.mowll radical distribution , eq 3d, We' obtain th e di sLribut ion of stable polymer by means of the relations : Combination:

Cit = n Ts( A )[k t( A ,A ).L;nik( A ) + k teA, B ).L;ntk(B )]+ n Ts( B)[k t( B , B ).L;n ik( B )+ lctC A, B ).L;n ik( A )]
(10 ) Equation 10 detl' l'mi nes t hl' instantanl'ou s di st ribution of sizes and composit ions in a ('opolyml'r formed from a monomer mixture of a given composition A j.A + B, whi ch , in tum, c1 etcrmin e Lh e values of w(A ) and weB ). If th e raLl' of th e reaction h as bel'n m easLl red, in tl'grat ion LllCn y ields th e toLal d isLribut ion obtained up to a given instant or degree of con version.
For the practicall y important case of long cha ins l'q 3d has bl'en simplified by Stockmayel' [28] in a manner analogous to that for simpl e polymers [12] . It will be noted that for t he la t ter case (s = O) , eq 3e reduces to the l'l's ul t obtainl'cl by Herington and Robertson [12] for th e distribution of radical lengths. Its physical significance is obvious. Sincl' w deviates from unity on ly slightly because of th e small concentration of radicals present, w T can, in a good approximation, be replaced for large r by e-( I -wIT . The final result is best expressed in terms of the number average chain length A of radicals, which eq ual s the ratio between the rate of reaction and t h e rate of production (or destruction) of radicals. Inserting this value and noting that the total concentration of rad icals is obtained from an equation analogous to eq 3b, one finds for the fraction of radicals of specified size [12]: To obtain th e corresponding expressions for copolymers we denote the total chain length r+s by l .
Deviations in the composition of a chain from the average value as given by eq 4a will be measurcn by the quantity ( 11) As we are concerned with large values of l, the sums in eq 3d may be approximated by integrals and the individual terms expressed by means of Stirling's formula. Considering that the d eviation from the average composition will not be very large in long chains, one can furthermor e expand the relevant expressions in terms of y . Th e final result may be expressed as a function of l and y [28] ( which gives the weight fraction of radicals with polymerization degrees between land I f dl and composition deviations between y and y + dy, irrespective of whether they terminate in employing the notation of eq 3b and noting that the factor in the bracket r epresents the total rate of destruction of radicals ' which equals the rate of production in the steady state. The first factor , the over-all rate of the reaction can be expressed as in eq 2a or by corresponding expressions for other r eaction mechanisms . It is assumed in eq 12 that monomers A and B have equal molecular weights.
The instantaneous size distribution of the stable polymer follows from eq 10. If termination occurs I by disproportionation, the weight fraction of copolymer in the specified range of land y is directly given by eq 12 . If a fraction p of the radical chains is terminated by combination, the weight fraction is obtained by multiplying eq 12 by a factor Thus in the approximations used to simplify eq 3d and granted the validity of th e reaction scheme 1 and of eq 4, the distribution function consists of two factors. One characterizes the distribution of molecular weights and the other the distribut ion of molecular composition for a fixed molecular we igh t. The specific nature of the tenrination process affects the former factor , but not th e composi tion distribution.
Finally, one derives from eq 12 the chain-length clistribu tion ilTespective of composition by integration over all compositions, ?a has the same form as the r esult obtained for the instantaneous distribution in pure polymers produced by disproportionation [12] . 'rhe distribution of composition flu ctuation s is ( 00 Plots of the expression (eq 12 ) as function of the reduced variables llA and ' YJ arc shown in figures 9 and] 0, respectively, ilS presented by Stoekmayl'l' [28]. It will be noted ,that large values of 71 , i, e. , large deviations from th e average ' composition, occur primarily in shorter chains, th e longer ch ains h av in g mainly t ll c avcragc composit ion . Filially, th c COl'llposition di strihution (eq 12b) is shown in figurc 11 foJ' c/ifrcl'cnt valu cs of the parameLer K. W ith in c'l'C'fls ing valuc of ft , th aL is, in creas in g valu cs of rr f.l for a fi,-:C'c/ Po, th c fluc t uatio ns in co mpos ition bccome largc!'. This is undcl'sLa ncl abh" s inc e in this in s tance self additi on of Aan cl B-units bccomes in cr cas ingly prdeI'J'C'CI . However, it will bc scc n th at th c devi ations from th e mean value y = O arc not la rgC'. Ac t ually it ca n be dcrived from cq 12 b til a t 88 percc'n t by wcight of th C' co polymer is found in' th e ra nge [y [< [2Po(l -PO)K / Aj" , which is small for la rge valu es of A. It should be notcd agai n th at application of thcse rcs ul ts to a completc copolymer product ]'cquires a knowledge of th e complete reaction mechanism which detcrmincs the variation of th e parameters A and K with avc ragc composition and conversion. Ko quantitat ivc data seem to be available. Fractionation of variou s copolymer system s has actually indicated t h e cx istence of a disp ers ion in respect to composit ion [13, 17 , 27].
The di scussion of eq 3d has sh own wllaLi s obvious, th a t eYrn for a fixed compos i ~io ll in a given c hain, thc rc a rc a vari ety of possi bl e in-ternaJ a rrangcmcnts of Lh c s pccics A a NI B in th c ('h a in corl'C'spo ndin g to seq uence's of ici cnL icaJ . Tll e w's h ave Lh c same meaning as in eq 3d , n amely , Lh aL of propagation probabili Lies A-A and R-R. Po . -Po If termination is effected by combination, th e dis-1.0 r ---------------------, tribu tion (eq 15) of sequences in the radi cal chains is not strictly id entical with t hat in the stable copolymer. Neglecting the effect of termination altogeth er , we simplify the equation for w(A) and weB) to: If the polymer in question h as been obtained for instance, by copolymerizing mono and divilwl units, eq 15 gives th e distribution of chain lengths between cross links and a number average ch ain length for a copolymer prepared from a monomer mixture con taining B moles of cross-linking agent.

Determination of Reactivity Ratios
The first t horough exp erimental investigation of copolymerization reactions h as been mad e by L ewis, Mayo, and Hulse [15 , 18], in which th e parameters lJ and jJ. were found for several pairs of monomers. As cq 6 cannot b e solved r eadily for lJ and jJ. , th e following procedure was adopted: In general these me thods do no t adequately separate the polymers, as shown by th e fact that th e r esults obtained by th e above workers [18] differed considerably, depending on the procedure used for the isolation of the polymer. Th e techniqu e finally developed utilizes the r elatively high vapor pressure of frozen ben zene at temperatures near 0 0 C. This method , known as th e froz en b enzene technique, involves several precipitations, after which the polymer is dissolved in benzene and then the solu tion quickly frozen. Subsequ ently th e benzen e is sublimed off under vacuum. The polymer is th en in the form of a very fine powder , which is easily handled. By using such means for the isolation of polymer , resul ts [15] were produced th at are very accurate for work of this nature. Almost all of t he monomer pairs so far studieJ have been those in which one monomer contained a different and easily analyzed clement or group _ In view of th e availabili ty and the hi gh dev elopment of spectrometric and other physical methods of analysis [9,16,19], there is room for techniques using these methods for the sLud y of copolymerization. Instead of analyzing the polymer , it should be poss ible to determine directly th e composition of the monomer residu e. One could place a sample of a polymerizing mixture in a high vacuum system and remove monomers from polymer by pumping off volatiles into a large residual volume or condensing in a liquid ail' tr ap. Then the volatiles could be analyzed by eith er mass, infrared, or ultraviolet spectrometry, depending on the nature of the system. An analysis of a relatively large bulk of residu al monomers should be less subj ect to errors due to small amounts of monomers trapped in th e polymer than an analysis of th e polymer itself at low conversion_ Such procedures should b e particula rly useful in copolymerization studies of isomers or compounds having small differences in their elemen tal analyses _ 2 _ Summary of Reactivity Ratios T able 2 summarizes the published results on monomer reactivities found by copolymerizing various pairs of monomers_ 5 It is seen, for instance, th at in t he reaction of styrene and methyl methacrylate, the addition of styrene monomer to styr en e radical occurs half as fast as t he addition of methyl methacrylate monomer to styrene radical. Also methyl methacrylate monomer adds half as fast to a methyl methacrylate ' W e are indebted to K. R . U en cry·Logan and R. V. V. N icholls for placin g t hcir rcsults at our disposa l prior to publication. The work was s ponsored by the Ollee of Ru bber R escr ve.  radical as does styr ene monom er. F ur th ermore, styr ene monomer adds to sty rene radical twice as fast as vinylidene chloride monomer to styrene radical. On t h e other hand , v in ylid en e chloride monomer adds to vin.ylid ene chloride radical approximately one seventh as fast as styr ene monomel'. The estimated degrees of precision are indica,tecl wh enever given by th e authors. In the case of references [3,5,7,38,39], the r eactivi ty ratios wer e obtained from plo ts according to eq 4a by flttin.g th e "best" curve. As an example, th e system styr ene-dichlorostyr ene investigated by Alfrey, M erz, and Mark [5] is shown in figure 13. A special case investigated is that wher ein one of the monomers does not polymerize with itself FW U R1, J 3. Plot of instantaneous concentrati on of dichlorotyrene i n polymer vs concentration of dichlorostYTene in monomeT according to eq 4a and da ta in [5] . mole ratio in monomer f 0 1" system styrene-diethyl ChlO1'Omaleate accol"ding to data in [5].

Journal of Research
but readily enters into copolymerization. Maleic anhydride [29] and its durivatives form a class of compounds exhibiting such a selectivity. Setting f.l. = 0 in th e first eq 4a results in a linear relation . An experimental test on the system styrene-diethyl chloromaleate is shown in figure  14 . The full line 0" = 2.5 is taken as the "best" value from a consideration of the data when plotted on a mole fraction basis. TIlE' foregoing results substantiate rather well the assumptions under which th e composition eq 4 has b een derived, at least for the relatively low degrees of conversion at which it may be expected to hold. In attempts to consider the effect of composition on the addition rates 6 equations have b een derived [20] for the case that the rates depend not only on the nature of the radical ends, but also on the preceding unit in the chain. The effects ar e small, and their detection would r equire consid erabl e experimental accuracy. Extensions of eq 4 to three-component systems and generalization to n-components [34] have been presented. Analysis of one four-component and seven three-component systems formed from styr ene, methyl methacrylate, acrylonitrile, and vinylidene chloride indicates agreement with I theory within experimental error. It is assumed in t his comparison that th e r eactivi ty ratios of a i pair A-B are independent of the medium and equal to the ratios obtained in the copolymerization of A and B alone.
Most of the published and analyzed results refer to bulk systems. In order to make valid comparisons for the monomer pairs studi ed, we shall confine the further discussion to th e data in table 2. Some investigations in emulsion sys tems have been und ertaken. For instance, the pairs given und er footnote 5 and also the styrene-acrylonitrile combina tion [15] have been analyzed both in bulk and in emulsion. In the main no significant differ ences between the 0" -f.l. values under the two different conditions are found.
, s e e also the d i'cu ssion ou rage 524.  [42] [ 5] [42] [        Tables 3 and 4 summarize the experimental information in a slightly different way by giving the relative reactivities of a series of different monomers vs a given radical, c. g., styrene. The scale is arbitrarily fixed by setting the reactivity of a monomer toward a radical of its own kind equal to unity. Values above unity then signify a higher activity than that exhibited by the radical in question toward the id entical monomer, whereas values below unity signify a lower activity. Some of the values may be affected by large errors, as can be judged by the limits givnn for the fJ and )L values from which they were calculated.

Referen ce
The interpretation of these results in terms of the electronic structure and internal geometry of radical and monomer is no easy task. However , certain qualitative attacks, at least, can be made It is perhaps worthwhile to prece de t he discussion of the subje ct propel' w it h a cursory and necessarily simplified summary of certain resul ts and concepts l"l'gardin g organic re actions. We need to conside]" th e influence certain substituent groups such as CH 3, Cl, N 0 2, etc., in vinyl-type monomers exer t on the elec troni c configuration in t he adj acent double bond. The presence of such groups in a benzelle r ing ll'ads to a ch ange, as comp ared with benzene, in th e r ates of further substitu tion and afreets also the locus of su bstitution. These facts have been known for som e time in class ical organic chemistry.7 They ar c caused by the tendency of an clectrophili c group to attack th e ring at the reg ion of hi ghest (relative) electr on density. Furthermore, if th e over-all density is redu ced compared to that in benzene, t he rate of reaction is accordingly reduced.
In considering the m echanisms of such distortions of the ch arge distrib u t ion , we shall somewhat arbitrarily separate two factors, inductive effects in the sense of Ingold 8 and resonance effects. TIlC fonner lead to an increase or decrease of the over-all availability of electron s in the riug, t.h e substituen t actin g as an electron source 01" sink . Fo r instance, a methyl g roup increases the electronic density in the ring and thus should incr ease the rate of substitution by an electrophilic agent while a chlorine atom, with its strong electron a ffinity, h as the opposite tenden cy. Similarly, substituents such as NOz, CN, or COOR tend to diminish the elec tron density . Analogous e.ffects by sll ch groups arc obser ved in addition r eactions to d Oli bJ e bonds in simple olefins. R esonance effec ts can determine the lo cus of attack:. For instance, group s with an unshared pair of electrons such as Cl, OH , N H z, C6H s, contribute struct ures to th e activated complex of the substitution r eaction (see footnot e 7) which make the ortho-and para-positions more n egative EIlC> X = _ _ e ~=C>, e while no stru ct ure mak ing the meta-position n egative can be wriLtell. Th e r esonance effect of N Oz, CN , COOR , and similar groups tends to leave th e m eta-position r elatively more negative. H ence they are m eta-directing. These directive efrects p ersist in addition reactions to olefins. In general, ortho-para-directing substituents cause addition to proceed in accordance with Markowni-kofI 's r ule, meta-directing ones in opposition to it. Th ese directive effects have also been explained w.ith out introdu cing explicitly the notion of r esonance [23J.
A quan titative meas ure of changes in electron density produced by substituen ts is given by Hamm ett's (J (sec footnote 8), constants. They are defined as t h e logari thm of the ratio of the ionization constan t of Lhe meta or para substituted b enzoic acid to t hat in the un substituted acid . A high positive value of Hammett's (J indicates a decr ease in elec tron den sity. The fundamental correctness of th ese concepts may b e judged from the results of certain calculations regarding th e charge distribution in substituted benzenes [37J.

. Induction and Polarization Effects and Relative Reactivities
Reverting now to the problem posed in the beginning, it may be seen that several factors ought to be considered in attempting to account for the relative r eactivities obtained and to predict the comparative behavior of monomers toward the same radical. The first ones arc the over-all availabilitv of electrons and, perhaps to a lesser extent, th~ direction of polarization of the double bond. L et us tak e first the styr ene radical as a basi for comparison. In the stable monomer , the double bond should have a relatively high electron density b ecause of the character of the phenyl group as an electron donor as evidenced by t h e direction of the dipole moment, which is opposite to that in toluene [45] . It should be rem arked, however , that Hammett's (J for the ph enyl groups is positive. Thus The data on the substituted styren es [35J in table 3 provide further support for the viewpoint expresse d above. They indicate the existence of a correlation between Hammett's (J values for the respective substituent group and the corresponding relative reactivities of the substituted styr ene monomers with styrene r adical. A high (J-value, which is characteristic, as said before, for a decreased electron availability on oth er groups attach ed to th e ring is accompanied by greater reactivity. A similar arrangement can be m ade in respect to the methyl m ethacrylate radical. Exceptions are encountered with resp ect to p-OCH3, and p-N (CH 3)2-styren e. The reactivities are high er than would correspond to the position of the substitu ent on the (J-scale. The authors [35] point out that these compounds are particularly effective in forming complexes with conjugated carbonyl systems. Generally su ch complexes can b e form ed by electron transfer between constitu ents of th e compl ex [36]. I n this connection the high selectivity ((J and /1-very small ) of allyl acetate and maleic anh ydride is of interest [7] . H ere th er e exists a possibility of resonance stru ctures b etween radical and monomer which corr espond to ch arge transfer within the activated complex. This possibility appears likely in v iew of th e colors formed by m aleic anhydride in mLxtures with electron donor-typ e molecules such as stilben e, indene, and styrene. The difference between maleic anhydride and a quinone-typ e inhibitor would be a matter of degree and d epend on t h e extent of reson ance stabili zation of the new radical formed . I nhibition is th en effected by th e removal of th e stable radical t hrough some furth er side reaction.
The direction and exten t of polarization of th e double bond can be of importance in favoring Journal of Res earch I j a high r eactiyity below A monom er such as the one should tend to react with a styr en e radical by adding h ead to 11 ead, wh ich would bring into play in terfer ence by th e group X , resulting in decreased activity as compared to th e case : aJ ,.e c lfF c l ) Consideration of the resonance stru ctm es in biphenyl (see footnote 8) suggests that the ph enyl group should br o-p -directing and th erefore polarize th e d ou ble bond in th e direction ind icated ~ by the firs t of th e above formulas.
The relat ive stabili t ies of the r adica l end s a re a second facior governin g tll e propagfl tion r a tes. The s tyr elle rfldi cal is rel at ively s table beca use of resona nce tllro ugh th e benzene ring. In cons idering t wo H'flctions, the one produ cing th e m ore stable radi cfl l end will be favor ed . Thu s in comparing vinyl ac etate wi th methyl acrylate monom ers, th e latter will exhibi t th e greate r act iv ity b ecause of ('onj ligation. The inclu ct i ve err('cts should be fl pproxim ately ('qu al in mag ni t ude for these t \\·o isom ers . R esona nc(' in L h e s ty rene raelical ,,·oul d be m ore impor ta nt LI mn in lIl e acrylate rad ical. T he valu e for til (' a('J"ylate monom er is neY(' rth eless sligh tly la rgCL" L1lUtl uni ty, probably , beca use of the d irection of the induct ive efrect away from the doubl e bond . One mi gh t expect the fo rm at ion of ac ryloniLril e radi cal to be somewllflt fa \~o re d by t he resonance e£-rect. F rom this point of vi ew all di enes should r eprese nt favorable cases .
vVe have been using for th e purpose of illus trat ion almost exclusiv ely th e styr ene radical because of the \\~e a l th of data available. The concepts developed are in fair agreement with experimental r esults obtained on oth er radicals. In th e case of v inylidene chloride , the radical reso nance s tabilization can play no significan t r ole. The chloride groups make the monomer positive in the sense preyiously discussed. T h erefore, it shows a high activity, with s tyrene monomet" and a lesser one wi th met ll yl m eth acrylate, whi e-h would not be extensively polarized bee-a usc of Lhe opposing eflects of th e m eLhyl and methyl substiLuted carboxyl groups. Wi th t he m ore positi vely pola1"-ized acryloni trile, the activity is still fur ther reduced . F or th e sam e r easons, a similar order of activit ies is obtain ed, at least in a quali tative way, for acryloni trile rad ical and m ethyl m ethacrylate radi cal with th e sam e m onomers. The electros tat ic efrects of g roups on cI iene radicals should be less impor ta nt tha n in t il eir m onom ers, becau se the ef}'ects would be m ost impor ta nt on the double bond, and the group is always ol1 e or mo re bond distances away from Lhe activated end in the case of l ,4-addit ion . The double bond i tself should act as an electron source. Actually Lh e fe w data ava ilable could be in terpreted by assuming that th e diene r adicals are som ewh at n egative r egardless of th e character of the s ubs t ituen t. H owcver , th e valu es for acryloni trile relat ive to the di enes do not appear to fi t t his picture.
TIl e mass of data given ca n be summ a rized in terms of cons ta nts that r d er to pa irs of r adicals and monom ers (sec table 2). It would be desirable ind eed to o bta in cons ta nts characteri s tic for each monomer as sllch . This would in principle all ow th e predi ction of reactiv ity rat ios . Th e complete reali zation of such a program seems remote i.n view of th e m any factors involved a nd Llw proba bl e ex istence of cou pli ng effects.

. Se miempirical relationships
Th e precedin g d iscuss ion makes it ev iden t th at th e polarity of rael ical and m onom er and the relative s tabi lit ies of the radicals arc th e m os t importan t facto rs Lo b c' consid ered . An atLem p t to find a set of cbaracteristic numbers in terms of th ese two eO'ects h as been recenLly m ade by Alfrey a nd P riCl' [6] . It is sugges ted Lh a t the various influ ences a rc separa ble and ca n be repr esented in the following way: wher e lc gB (A ) is th e rate of addit ion of monomer B to ra dical A . P A is characte rist ic for the rad ica l. T he e's are a meas ure of the cffective cllflrge on Lhe end of th e radical taken to be i([e ntie-al with the charge on th e dou ble bond of mo nomc r .It an d on th e double bond of mo nom er If, r es pectively. QB r epresen ts a 111ean r eactivi ty of mo nomer B obta ined by fo rm ing th e geonll't ri c mea n of the r l'acLiviLic's of B with a seri l's of radi cals A , B, . . . and Llw il ass igni ng to one monomer a refer-e11("e value of uni ty. T il is equa tion implies that Uw free energy of activation for t ll c' propagation step is additive in respect to t he above-named effects. In comparing the behavior of monomers A and B toward radical A, P A cancels out and one obtains for the ratios :  Some difficulties arise if one attempts to fi t th e other olefins studied in to the fram ework represented by tabJe 5. Th e Q-values are no t always consistent with th e interpretation given to them and neith er arc th e polarities. No satisfactory r esults are obtained by applying eq 17 and 18 to the data on dienes obtained by H enery-Logan and N icholls (see footno te 5). In deriving these results the equali ty of effective charges for monomer and radical is assumed. As m en tioned previously , it seems to us that in th e case of clienes a differentia tion ough t to be made particularly when th e inductive effect of a substituen t acts to decrease the electron availability [32]. On t his basis one can evaluate th e pertinent data in conjunction with th e values for the vinyl compounds 540 given in table 5. We modify the two eq 17 and 18 by replacing the first factors eA and en by e*4. and e*n and one factor (eA -en) in eq 18 by (e*A -e*n) , respectively . The star refers to the radical. The result of th ese calculations is summarized in table 6. The differences in sign between the e-values are in the expected direction. All e*-numbers are negative, th e one for ehlO1'oprene being the least. The few data available for an independent ch eck are rather well reproduced by using table 6. It can be seen from eq 17 that a large valu e of th e ratio QA / Qn for a set of e's corresponds to large values of u and small values of J.I. . Thus monomers with great disparity in th e Q's, will copolym erize poorly. Large differences in the e's, of course, lead to good copolymerization.

. Steric Effects
In considering the copolymerization of compound s such as stilbene, indene, m aleic anhydride, the mal eates, and the fumarates, another factor becomes important, n amely steric hindrance. For example, the symmetrical substitution of another phenyl group in styrene leads to a comp ound, stilbene, which does not polymerize. Maleic anhydride and other symmetrical disubstituted ethylene deriv atives polymerize with difficulty, if at all. However , stilben e and m aleic anhydride form copolymers. The hindrance in such a case should still be great but is apparently overcom e by th e influence of polarity effects since t he pertinent double bond can be expected to be positive in maleic anhydride and negative in stilbene .
A striking example of steric hindrance is provided by the comparison of maleic anhydride, diethyl chloromaleate, and dicthyl maleate 14, 5] .
The reactivities with styrene radical are respect ively 24, 0.4, and 0.2. Th ere differences h av e been ascribed to the opening of the anhydride ring [24]. D iethyl fumarate h as a reactivity of 2.5 toward styrene monomer [24]. The increase over that of its cis-isomer can be understood on the basis of the geom etry of the two mol ecules, if the respective resonance stru ctures of maleates and fumarates ar e considered.
The preceding discussion dealt entirely with r elative rates of propagation in copolymerization, which are th e ones determining average composition. Differences between monomers are, of course, to be expected in r espect to the other steps of the ch a in reaction. In considerin g for instance , t he rate of peroxide induced initiation, the polarities of monomer and catalyst radical and the stabilities of the radicals formed are of importance. We would expect the phenyl and the benzoyl r adi cals to be mgative. It is not poss ible to compare directly the rates of initiation in two binary systems, since the rate of decomposition of the peroxide depends markedly on the rnedium [22]. This can be minimized by using dilute solutions of thc monomers in an identical solve nt. It serves no purpose to discuss in any greater detail poss ible effects of monomer sLrucLul'e on the relative rates of elementary acts other than propagation , until a complete kinetic analysis of th e copolymeri zation of at least some typ ical pairs has been obtained.

. Effect of intramolecular Arrangement on Degradation
The whole diseussion has hiLherLo rdel'l'ed to the building up of copolymer chains. It is of in terest to consider also the reverse process. It is not our purposc here Lo di scuss in dctail thermal decomposition of polymers. IVe mcrely wish to point out briefly th e relationship between th e stru cture of the copolymer as considered previously and the res ul ts to be ex pected in iLs degradation. Studies of the thermal decomposition of various copolymers have shown that in many cases th e yield of monomers are mu ch lower than what would be expected from the number of monomer units known to be in the polymer and the behavior of the simple polymer [33]. For example, the yield of styrene from GR-S is much less than tbe eOJ'J'esponding yield of styrene from polystyrene.
Assuming that th e effect of side reactions on th e yield of a given monomer rcmains consta nt in going from simple polymer to copolymer depolymerizations, the pyrolysis yield of certain types of monomers obtained from a copolymf'r of given composition may be calculated. L et A represent monomers of Lhe mono or asymmetrical ciisubsti-Copolymerization tu ted eLhylene type (CH2CXY) with h ead-to-tail arrangement in the simple chain and in th e copolymer, and B monomer of the diene of symm eLrically disubsti tu ted ethylene types. In comparing th e expected yicld of A from a copolymer with the co rrespo ndin g one from a pure polymer A, wo proceed in th e followin g manner. Consider a seq uence of i A-uniLs, wh ich in t he case of th e copolymer is bounded by B uniLs. There ar e 2i possibilities of produci ng a split, 2i-l in the interior and 2 X % at the boundaries, where the factor % is included to avoid twofold cou nting of the bonds joining th e sequence to th e resL of th e chain. 2i-l of these spliLs prod uce monomer A. If the probabiliLy of OCC lll'l'ence of a sequence of length i is clenoLecl uy P i(A) (eq 15), th e yield Y c of A from the copolym er becom e equal to: (19) where Yo denotes the expected yield of monomer from Lb e pme polymer A . In deriving eq 19, it has been assumed that splittin g occurs at random and independent of the nature of the adjacent unit in th e chain. Also recombination is excluded. One would expecL large positive cleviaLions from the calcu lated yield s Lo be an indication of head-tohead and Lail-Lo-Lail sLructul'es in th e polymer. For in sllch a case, th e effect of the sequen ce boundaries considered in eq 19 is absent. In practice, the number of sllch configuraLion is us ually small a,nd obscured by oth er factors. Using the exp ressions (eq 15 apd ] 6), \\'e finally obtain from eq 19: (19a) Since t h e composition of tbe copolymer can be determined from eq 4a, the thermal decomposition yield can be plotted against Lhe instantaneous polymer composition as shown in figure 15 for the styrene-butadiene system using the pertinent values of 11' and jJ.. For certain copolymers, such as the polybutenes, where the monome rs arc isomers 01' otherwise similar, this ma,y be a useful tool for the devclopmen t of a pyrolytic analyLical techniqlle, particularly sincc most monomcrs can be dctermined spectromeLrically, ,,·hile the copolymrrs cannot always be analyzed . The full appliea,t ion of this technique A . E xperimental for G R -S.
will req uil'e precise con tl'ol of pyrolysis condi tions and highly refined analytical methods . At present, t h e above concepts will account for some of the results found in the depolymerization of copolymers. 6 . Conclusion In conclusion, the obvious main lines for further work may be sketch ed. Kinetic studies on copolym ers are few and will have to bc extended to includc over-all rates, and at least in some typical eascs, determinations of the individual rate constants of the chain reaction. It will be further interesting to see whether the observed r eaetivity ratios can be r elated to the structure of the monomer through other constants characteristic of the same. Considcration of dipole moments, polarizabjJities, and ionization potcntials may offer some clues. Indications as to steric influences in the copolymer ch ain may be gained by a comparison of the heats of reaction of the pure and mixed species. 9 The preceding discussion dealt primarily with the mechanism of form ation and the resulting structure of th e copolymer. Apart from the work on GR-S, no systematic studies of the relation between these factors and the thermodynamic [26] and rate properties of copolymer solutions seem to have been und ertaken. Some physical properties of 9 See in this connection a remark made by l\1. O. Evans, J . Chcm. Soc., 1941 .264. L. K. J. Tong and \\" . O. Kenyon, 11 3th meetin g of the American Chemical Society, Chicago. llIoO April 19 to 23.1948 542 certain copolymers besides synthetic rubbers have been systematically investiga ted. 10 VI. References