3 EMULSION
POLYMERIZATION
A. E. ALEXANDER* and D. H. N A P P E R t Department oJ'Physical Chemistry, Unirersity of Sydney. N.S.W. 2006, Australia CONTENTS List of Symbols I. Introduction 2. Theory 2.1. Interval 1 2.2. Interval II 2.2. I. The SmithEwart theory 2.2.1.1. Case2 2.2.1.2. Case3 2.2.2. The StockmayerO'Toole theory 2.2.3. The Gardon theory 2.3. Intervallll 3. Experimental 3.1. Rates of Decomposition of Initiators 3.2. Entry of Free Radicals into Particles and Micelles 3.3. Number of Particles in Interval I 3.4. Number of Particles in Interval I 1 3.5. Interval II Kinetics 3.6. Mechanism of Termination 3.7. Poorly Watersoluble Monomers 4. Colloid Stability 5. Conclusions 6. References
145 147 148 148 155 157 158 158 159 164 172 17~ 173 176 1~ i 1~3 187 188 189 191 194 195
LIST OF SYMBOI.S
A' A sE
A~, a a.~ a,,,
effective H a m a k e r c o n s t a n t , erg S m i t h  E w a r t case 3 p o l y m e r i z a t i o n v o l u m e rate, I. (I. o f emulsion)' a r e a o f l a t e x particles per unit volume, c m z (1. of emulsion) 1 ( 8 v ~ , f N a / r k t ) '1~ area occupied by unit weight of surfactant, cmeg a r e a o c c u p i e d b y s u r f a c t a n t in o n e m i c e l l e , c m e
B sE
SmithEwart s i o n ) ~
case 2 polymerization
Cu
m o n o m e r c o n c e n t r a t i o n in latex particles, mole (I. particles)
*Deceased May 23, 1970. fQueen Elizabeth 11 Fellow. 145
v o l u m e r a t e , 1. (I. o f e m u l 
146
A.E.
A L E X A N D E R A N D D. H. N A P P E R
Cn concentration of free radicals in water, moles (l. of emulsion) 1 D di dp f f, H0 I~ k~ ks
diffusion coefficient of free radicals, cm 2 sec 1 density of m o n o m e r , g cm 3 density of dry polymer, g c m :~ volume factor allowing for swelling of p o l y m e r by m o n o m e r fraction of particles containing n radicals distance of closest approach of two spheres, cm modified Bessel function of first kind of order ~, propagation rate constant, I. mole ~ sec 1 radical exit rate constant, m m mole 1 sec 1 kt radical mutual termination rate constant, 1. mole ~ sec ~ L n u m b e r of lattice sites per latex particle [M] overall m o n o m e r concentration, mole (i. of emulsion) 1 [M]0 initial overall m o n o m e r concentration, mole (l. of emulsion) 1 Mw molecular weight of m o n o m e r , g mole ~ (MN) n u m b e r average molecular weight, g mole ~ Minst instantaneous average molecular weight, g mole 1
m
(kss/kt)
N
n u m b e r concentration of latex particles, 1.1 A v o g a d r o ' s constant, mole ~ n u m b e r concentration of micelles plus particles, 1.1 n u m b e r concentration of particles with n radicals, 1.~ n u m b e r concentration of dead particles, I.~ n u m b e r concentration of growing particles, 1.1 n u m b e r concentration ofmicelles, 1.1 n u m b e r concentration of latex particles at time t, 1.~ n u m b e r of free radicals in a particle average n u m b e r of free radicals per particle total n u m b e r concentration of free radicals, 1.~ the order of magnitude free radical concentration, mole (1. of emulsion ~) radius of a spherical particle, cm centretocentre distance of two spheres, cm weight of soap per unit volume, g 1.1 surface area of a swollen latex particle, m '~
NA Nr
N, Na Nf/ N,,,
Nt n nr 0 [R] r R S s
s*
(R/r)
t time, sec tM average time between successive m o n o m e r additions, sec tcr duration of interval I, sec VA attractive potential energy, erg Vn repulsive potential energy, erg Vr total potential energy, erg Vp volume concentration of unswollen polymer, I. (l. of emulsion) 1
EMULSION POLYMERIZATION V~:
Fv v~ vj, x ;: z* cl~
~3' y* i; 1/K A t.t q' p t)' p* ~r r r :~ (h~
147
volume of one m o n o m e r molecule. I. partial specific volume of m o n o m e r , cm :~g 1 volume of swollen p o l y m e r particle, I. volume of an unswollen p o l y m e r particle, 1. dimensionless p a r a m e t e r proportional to time dimensionless p a r a m e t e r proportional to conversion subdivision factor (NAv'~/k:)
dielectric constant particle radical capture efficiency factor micelle radical capture efficiency factor ~potential, volts D e b y e l e n g t h , nm average n u m b e r of terminated molecules per particle particle volume increase per unit time, cm a s e c  ' dimensionless p a r a m e t e r of O(kt/kt,). rate of generation of radicals in aqueous phase, I.~ sec rate of entry of flee radicals into latex particles, 1. ' s e c ' rate of entry of radicals into micelles plus particles, I. ~ sec ~ second neighbour coordination n u m b e r interval between successive radical entrances, sec time of nucleation of a particle, sec volume fraction of m o n o m e r
1. I N T R O D U C T I O N T h e literature of emulsion polymerization contains a vast array of experimental and theoretical studies. Because of the complexity of the processes involved in this type of polymerization, many of the experimental studies have yielded what are seemingly divergent results. N o t surprisingly, the conclusions drawn from these studies are often contradictory. Little purpose would therefore be served by presenting here an uncritical catalogue of the more recent studies of emulsion polymerization. The experimental approach to emulsion polymerization has latterly b e c o m e s o m e w h a t stereotyped. This has led to a decline in the execution of critical, definitive experiments which are vital to the further progress of what is a complex, and industrially very important, method of performing polymerizations. This review aims at examining in some depth selected papers which, in the authors' opinion, have contributed in an important and novel fashion to the discussion of the m e c h a n i s m of emulsion polymerization. N e c e s sarily, what is important, and to some degree w'hat is novel, is a subjective
148
A.E. ALEXANDER AND D. H. NAPPER
matter; at worst, however, this subjective approach seems to be the lesser of two evils in such a diffuse field. Several excellent reviews of earlier studies already exist. ~14~The reader is referred to these for discussions of the many facets of emulsion polymerization which are not touched upon here. N o apology, however, is needed for the deliberate bias in this article towards a consideration of the surface and colloid chemical aspects of emulsion polymerization. Their importance, in the past, has not perhaps always been fully recognized. T o redress this omission, it has been necessary to curtail, or excise completely, consideration of certain important features of emulsion polymerization for which adequate discussions already exist, or for which no significantly new ideas have appeared recently. The p h e n o m e n o n to be considered is the isothermal emulsion polymerization of a single monomer, which is capable of swelling its waterinsoluble polymer. T h e initial reaction charge consists of emulsified droplets of monomer, and surfactant in excess of its critical micellar concentration (c.m.c.). A watersoluble initiator is added and polymerization proceeds, generating a stable latex. Despite certain imperfections and obscurities, the general mechanism first propounded by Harkins ¢5'6~ and others ~1"7~ remains the one which has achieved the most widespread acceptance. T h e broad features of this picture will therefore be retained in this discussion. The different reaction intervals will be designated by roman numerals ~8~to distinguish them from the various cases (denoted by arabic numerals) considered in S m i t h  E w a r t theory. ~9~
2. THEORY 2.1.
Interval I
Interval I begins with the onset of free radical generation and proceeds until all the micelles are consumed. Often only 1015%, or even less, of the total m o n o m e r present is polymerized during this period. This small conversion in no way diminishes the importance of interval I to the overall theory of emulsion polymerization since it appears that, during this interval, all the latex particles are generated. The number of latex particles ( N ) so formed may significan;l!ly influence the postmicellar course of polymerization in intervals II and Ill. The theory of interval l therefore centres on the calculation of N and the rate of polymerization. A normal emulsion charge, before initiation begins, consists of emulsified droplets of monomer (~> c a . 1/~ diameter, thus giving 101710 TM droplets/1, of emulsion), m o n o m e r solubilized in micelles (usually there are 10a91022 micelles/1, of emulsion) and m o n o m e r dissolved in the aqueous nonmicellar regions. Free radicals are usually generated in these
EMULSION POLYMERIZATION
149
nonmicellar zones at a constant rate (p) and enter the micelles (and the particles it" present) at a constant rate p * ( p * <~ p ) . Particle nucleation is assumed to occur when a free radical enters a micelle: these particles grow unless terminated. T h e emulsion droplets apparently make a negligible contribution to the overall rate. due to their small number. Polymerization in the aqueous nonmicellar regions is likewise regarded as negligible, at least for sparingly watersoluble m o n o m e r s . Smith and Ewart (~') were the first to formulate theoretically an upper and a lower limit to the n u m b e r of particles generated in interval 1. This number is considerably fewer than the n u m b e r of micelles originalb, pre,,;ent, usually being in the range 10t'~10 ~s particles/l, latex. Their predictions rested i n t e r alia on three basic assumptions: first, that the interfacial area (a~) occupied by unit weight of surfactant is the same for the micelles and the soap saturated polymer particles: second, that the ordinary laws of diffusion are operative in the uptake of fl'ee radicals by the micelles and the latex particles: third, that the rate of polymerization is constant in the p o l y m e r particles once nucleated. T h e critical micelle concentration (c.m.c.) of the surfactant and the amount of surfactant required to stabilize the emulsion droplets are both assumed to be negligibly small. Smith and Ewarg '~) derived one value for ,,\" by postulating that all the free radicals enter the micelles. This yields an upper limit, as some flee radicals actually enter the p o l y m e r particles. T h e constant rate of increase of the volume of each pm'ticle (p,) implies that the area of a single particle (a~ ,r), which was nucleated at time, 7 :~, is given at a subsequent time t by a<., =
4n) i.~ 3
# d r '~: ~r
= O ( t   r * ) "'/:~
(1)
where 0 = [ (47r) ~2 3/x]2/:L T h e total area of p o l y m e r (A,) per unit volume of emulsion at time t is therefore A,, = p~: f ' a_ .,d~*
= 0.60 p*OtS/:L
(2a) (2b)
N o micelles remain when A p = a~S. where S is the weight concentration of surfactant. Thus interval I is complete after a critical time t,,,. = (a.~S/O • 60 p*0) 3/5 = 0 • 53 (a~S/p'~) a/5 (1//x) "/'~. An upper limit for N is accordingly N = p*t~.r = 0'53 (p*/lX) ''/5 (a~S) :~/5.
(3)
150
A. E. A L E X A N D E R A N D D. H. NAPPER
T o calculate a lower limit, Smith and Ewart ~ assumed that the effectiveness of radical capture per unit surface area was independent of the size of the species on which that area was located. Simple diffusion theory predicts that micelles have a greater capture efficiency per unit area than do latex particles. <1°12~ This assumption therefore biases radical entry towards the particles. T h e rate of particle nucleation is obtained by decreasing p* by that fraction of the free radicals which enter the latex particles: d N / d t = p* (1  [A,,/a~S] ). (4) F r o m eqn. (2a)
d N / d t = p*  (p*O/a,.S) f~ ( t   T * ) z/3 (dN/dT*)dT*.
(5)
An approximate solution of this i0tegral equation, which is in the form of a Volterra equation of the second kind, is N = 0.370 (P*ltz) '2/~ (asS) 3/~.
(6)
G a r d o n ~8/has confirmed the correctness of the S m i t h  E w a r t solution (6) by a more exact numerical analysis. Comparison of the upper and lower limits for N shows that
N = k(p*/I.t) 2/5 (asS) 3/~
(7)
where 0.37 < k < 0.53. Thus N is usually taken to vary with the 0.6 power of the initial soap concentration and with the 04 power of the initiator concentration. G a r d o n (8~ has recently argued that the S m i t h  E w a r t lower limit in fact predicts the exact value of N. This presupposes that during interval I, free radicals are captured by particles and micelles at a rate which is proportional to their surface area. T h e justification of this proposition rests upon a geometrical analysis of the probability that a radical formed in the aqueous phase hits a particle. Essentially this analysis reduces to the statement that the collision crosssection of a latex particle of radius r is proportional to its maximum crosssection rrr 2, which is onequarter of its surface area. No allowance is made in this model, however, for the free radical concentration gradient which surrounds the particle as a consequenc¢ of the zero radical concentration presumed to occur at the particle surface, acting as a free radical sink. Not surprisingly, the result of G a r d o n ' s geometrical model is at odds with the prediction of the ordinary laws of diffusion. Von Smoluchowski (1°12~ calculated the rate of fast coagulation (i.e. in the absence of a repulsive potential energy barrier) of colloidal particles
EMULSION POLYMERIZATION
15 I
when their (London dispersion) attraction is approximated by an infinitely deep potential energy well, with a vertical wall, at their distance of interaction. This ~theory suggests that the rate of entry of uncharged oligomeric free radicals into a latex particle may be approximated by 47rDrC~,where CR is the maximum free radical concentration in the aqueous phase and D is the free radical diffusion coefficient. Thus simple diffusion theory demands that the rate of capture be proportional to the first power of r and not to the second; accordingly, the free radical flux through unit area would be expected to be greater for smaller particles. The yon Smoluchowski equation has been shown to predict, within a factor of two, the rate of last coagulation of uncharged colloidal particles. {~a~ H o w e v e r , this relation is an approximation in so far as the dispersion attraction is not infinitely large but is a function of the particle size ~14~ and may extend appreciable distances (i.e. several nanometres) into the aqueous phase. F o r the entry of charged fl'ee radicals into charged pm'ticles, allowance also must be made for electrostatic interactions, perhaps along the lines suggested by Fuchs ~a~ o r . D e b y e J ~6~The question of the dependence of the rate of entry of free radicals upon particle radius is thus still an open and complex one, particularly if electrostatic interactions are involved. [t will be shown in Section 2.2.1.1 that, under certain conditions, the rate of polymerization in interval II theoretically may be constant, corresponding to an average number of free radicals per particle (~) of 0.5. The assumptions made by Smith and Ewart in their upper limit calculations of N obviously predict that ~ is 1.0 at the completion of interval 1: the value of h for the lower limit assumptions was not calculated. The recent numerical analysis by G a r d o n {~ for the lower limit case gives ~ ~ 0.67. Thus at the completion of interval 1, the S m i t h  E w a r t assumptions predict that 0.67 < t~ < 1.0. Clearly h is considerably larger at the end of interval 1 than the value of 0.5, often considered to hold for interval I1. Parts and Moore ~r~ and van der Hofla'~ concluded that the SmithEwart assumptions presage a maximum in the rate of polymerization at the time of disappearance of the micelles. This conclusion is valid only if the value of~ in interval 11 is restricted to ca. 0.5. N o maximum in the rate would be anticipated if ~ is significantly larger than 05 but, if this were so, the rate of polymerization should presumably increase throughout interval 11. In general, the most careful rate determinations ~ do not exhibit the expected maximum at the requisite point. This implies either that the assumptions adopted in the S m i t h  E w a r t derivation are incorrect or that h is significantly greater than 0.5 in interval 11. The constant rate of polymerization sometimes observed in interval I1 suggests that, in these instances at least, the former inference is correct. G a r d o n ~s~has proposed another explanation for this anomaly: because the excess rate at the end
152
A.E.
ALEXANDER AND D. H. NAPPER
of interval I, o v e r the subsequent rate in interval 11, may only be ca. 35% (incorrectly given as 26%), the predicted maximum is experimentally undetectable. Since rates of polymerization are measurable to perhaps an accuracy of 5%, this explanation demands that the period of rate excess be smaller than the response time of the various methods used for monitoring the rate. A numerical analysis suggests that this is physically unlikely/1.~ Parts. Moore and Watterson ¢~ have endeavoured to extend the SmithEwart approach by proceeding more rigorously from the hypotheses of H arkins) :''6~ The additional assumption invoked was that mutual termination of free radicals in the polymer particles is instantaneous. The total number of particles (Nr) into which the radicals may enter is the sum of the number of micelles (N,,,). and the number of growing (N,,) and dead (Nd) latex particles
Nr = Nm + N~j + N,t.
(8)
Because of the gross uncertainties in the relative radical capture efficiences of micelles and particles, the simplest assumption to make is that the probability of radical entry into a given species is proportional to the number of such species. Consideration of the ways by which the different species are formed or removed shows that
 (dN,,,/dt) = p* (N,,,/NT) + Q (t),
(9)
(dN,/dt) = p*( [N,,, + Na N , ] / N r ) ,
(10)
(dNa/dt) = p*( [ N , ,  Na]/Xr)
(I I)
and hence
(dNr/dt) =  Q ( t ) .
i12)
In eqn. (9), Q(t) denotes the rate of disappearance ofmicelles resulting from the adsorption of the surfactant onto the surface of the latex particles. For a constant increase in the volume of latex particles, Q(t) would be proportional to F ~3, if no dead particles were formed. Because dead particles are formed, Q(t) depends upon a smaller power o f t ; it is such a complicated dependence, however, that a simple solution of this set of differential equations is not yet possible. If Q(t) is presumed to be constant (=)t) instead, the set of differential equations can be solved readily. The fraction of particles which contains one radical when interval 1 is complete is
N¢,/Nr= (p*/[A 2p*])[(X/p*) I[22p*]/fap*]) 1].
(131
EMULSION POLYMERIZATION
153
This relation was applied to the experimental results of Bartholom6 et al., ~ls~ which enable ~ to be calculated from the experimentally determined time of micelle disappearance. A typical value for N~/NT is 0.98. i.e. only 2% of the latex particles existing at the end of inverval I have captured a second radical. T h e assumption that 7, is constant gives a lower limit to the ratio N J N ~ since the rate of disappearance of micelles in reality increases as interval 1 proceeds. Thus N,,]Nr ~ 1.0, even in its lower limit. Allowance for the differences in micelle and particle sizes may be introduced by adopting the von Smoluchowski rate law. The rate of increase in surface area of a spherical particle which increases its volume at a constant rate p,, is simply (2p,/r). The new set of differential equations is ~.* = Nmrm + N~,(r~j) + Na(ra) (14)   ( d N , , / d t ) = p* N,,r,,,/E * + 21~N~j/a,,(r,,),
(15)
(dNf~/dt) = p * ( [ N m r , , + N ~ j ( r a )  N~(r~)]/'£*),
(16)
( d N a / d t ) = p* ([N~,(r~)  Na(ra) ]/~ * ).
117)
H e r e (r) denotes the appropriate instantaneous average of the particle radius, and a,, is the total area on the surface of a latex particle covered by all the soap molecules constituting one micelle. This set of differential equations is most readily solved with the aid of an analogue computer, given appropriate numerical values for the variables. Parts, Moore and Watterson ~ obtained their solution by alternative numerical methods, and found once again that N~]Nt ~ 1 at the end of interval I. The reason for this in all instances is that the number of micelles present greatly exceeds (e.g. by more than several orders of magnitude) the number of latex particles. This situation persists right up until the moment when almost no micelles (< 1% of those initially present) remain. If h = 0.5 in interval II, these calculations show that a maximum in the rate of polymerization must result, given the S m i t h  E w a r t postulates. Agreement between theory and experiment can be achieved, however, if an assumption indicating preferential radical capture by the latex particles is introduced. A radical generated in the aqueous phase thus bears a different relationship to already formed latex particles from that which it bears to the micelles. This bias was introduced mathematically into eqns. (14)(17) by replacing the product term N,,r,,, by y*Nmr,,. where y* < I. When a value of 3'* = 001 was employed, for example. N~,]NT was reduced from ca. 1 to 0.516. N o maximum in the rate curve is
154
A. E. A L E X A N D E R A N D D. H. N A P P E R
then expected to be detected by conventional measurements, in agreement with experiment. No justification for 3'* in physicochemical terms was advanced, although several possibilities may be suggested. For example, the London dispersion attraction between particles and free radicals would be greater than that between micelles and free radicals, because of the larger size of the latex particles. ('4) Electrostatic interactions may also influence differently the entry of charged free radicals into particles and m~icelles through differences in the surface charge density. Neither of these effects are anticipated to reduce 3'* to 001, however, so that other factors, such as the relative rates of oligomer incorporation, must be invoked. The parameters of interval I theory which are often tested experimentally are the 0.6 exponent of initial soap concentration and the 04 exponent of the initiator concentration. In principle, the rate of polymerization in this interval constitutes a third verifiable parameter. It will be shown in Section 2.2 that the instantaneous rate of polymerization in interval I is given by  ( d [ M ] / d t ) = (kj,/NA)CMhN t.
(18)
Here Nt is the number concentration of latex particles at time t and CM is the saturation concentration of monomer in the latex particles. The upper limit assumptions of Smith and Ewart ¢9)yield Nt = p ' t , since each particle contains one free radical. The integrated rate equation, which is the form most suited for comparison with experiment, is thus Vp = 05 (k,/NA) (d~/dp)qSMp*t",
(19)
where V~ is the volume concentration ofunswollen polymer. Ideal mixing of monomer and polymer has been assumed in this derivation; if mixing is not ideal, dM must be replaced by 1/~M. Equation (19) holds over the entire period of interval I, subject only to the lower limit assumptions. Gardon (s) has shown by a numerical analysis that the upper limit SmithEwart assumptions yield Vp = 035(1  qSM)/zp*t2 [1 0.69tz°'6Z(p*/asS)°'9'~t"~ ] .
(20a)
Equation (20a) is valid over the entire interval I but assumes ideal mixing of monomer and polymer. For nonideality, the factor (1 ~bM) must be replaced by the reciprocal of the particle swelling factorf. For short times (t ~< 05 tcr), only the first term in brackets in eqn. (20a) is significant, so that VI, = 035 (k,~/NA) (di/d,,) qSip*t o
(20b)
where/z has been replaced by (kp/Na) (dM/de)qSg/( 1 OHM). The upper and lower limit SmithEwart assumptions ought to yield an
EMULSION POLYMERIZATION
155
upper and a lower limit to the rate of polymerization. Hence, in interwtl I, if t ~< 0.5 t~.r, V~, = k' ( k , / N A) (du/d,,)dJ,up *t ~ (21) where 0.35 < k' < 0.50. A plot of the square root of the conversion is thus predicted to be linear with time, provided that mutual termination of free radicals in the polymer particles is instantaneous (ct'. a similar prediction for interval I1 if termination is not instantaneous). The particle size distribution and the molecular weight of the t\~rmed polymer constitute two more experimentally observable parameters in interval i. Both are intimately associated with the relative capture efficiencies of micelles and polymer particles. Until this problem is solved, quantitative theories of particle size distribution and molecular weight distribution must remain somewhat speculative. Watterson, Parts and Moore ¢~°' have derived expressions for the integral average degree of polymerization of the polymer formed in both intervals 1 and 11. This derivation rests on the assumptions adopted by Smith and Ewart in their upper limit of N calculation: it requires, in addition, that transfer reactions be negligible and that mutual termination of fl'ee radicals in polymer particles occur instantaneously. The main conclusion of this analysis is that there is a discontinuous increase in the integral average degree of polymerization on completion of interval 1. This discontinuity is associated with the maximum which is predicted, on the basis of the same assumptions, for the rate of polymerization at the same transition point. The latter has not been observed experimentally nor, apparently, has the molecular weight discontinuity. G a r d o n ~) was unable to estimate the interval I molecular weight with any certainty when the lower limit S m i t h  E w a r t assumptions were adopted. A less pronounced discontinuity is anticipated in this case. 2.2.
Interval II
Interval 11 begins immediately all the micelles disappear. It is assumed that once this happens, particle nucleation ceases and the particles formed in interval I grow without flocculation. Interval 11 is accordingly characterized by the constant number of latex particles (N) formed in interval 1. These particles, which are swollen with monomer, are considered to be the sole loci of polymerization. T h e full kinetic effect of compartmentalization of the free radicals into a large number of latex particles can thus manifest itself. The diffusion of monomer from the remaining emulsion droplets into the particles is postulated to be sufficiently rapid to maintain equilibrium saturation swelling. The concentration (C,~z) of monomer in the latex particles is therefore constant in this interval. Interval I1 is
156
h.
E. A L E X A N D E R
AND
D. H. N A P P E R
complete when no monomer droplets remain. The lower bound of this interval is therefore the disappearance of the micelles, while the upper bound is the disappearance of emulsion droplets. During this period, perhaps onethird of the monomer is polymerized. Free radicals are generated at a constant rate (p) in the aqueous dispersion medium and enter the particles one at a time, at a constant average rate (p', where p' = yp and 3' < 1). The capture efficiency term (y) is necessary to allow, for example, for the possibility of different rates of entry of oligomeric free radicals into particles whose colloid stability is imparted by different mechanisms. The average interval (r) between successive radical entrances is N/p'. The particles are assumed to be monodisperse throughout the interval; no allowance need thus be made for different rates of radical capture by particles of different size. The amount of monomer which is polymerized onto the primary free radicals prior to their entry into the particles is presumed to be small. Mutual termination is postulated to be the sole mechanism responsible for free radical annihilation in the latex particles. Let the system consist of No, N1, N2 . . . . . N , . . . particles containing 0, 1,2 . . . . . n . . . free radicals. The overall rate of polymerization is obtained by summing the individual rates of polymerization in the different types of particles: ~e
d[M]/dt=
•
(k~/NA)CMnN,.
(22)
n=0
The calculation of the rate of polymerization in this form requires knowledge of the normalized free radical distribution function. To circumvent this requirement, the problem is often posed, and solved, in a somewhat different guise. The known value of the total number of particles is simply oo
N = Z N,
(23)
n=0
whilst the total number of free radicals (nr) per unit volume of dispersion is nT= ~ nN , . (24) cc
n=0
The average number of free radicals per particle (h) is given by h = nv/N = ~ n N , / N .
(25)
11=0
Insertion of relation (25) into eqn. (22) yields   d [ M ] / d t = (ko/Na)CMN~.
(26)
EMULSION
POLYMERIZATION
1 5"7
T h e problem has now been transformed into the calculation o f ~ instead of the free radical distribution function. Most calculations todate of h rest on the application to interval 11 of the steady state condition. This was lirst examined in detail by Smith and Ewart e'' and later by H a w a r d : ~1' T h e steady state condition implies that the rate of generation of N,,type particles can be equated to their rate of disappearance. N.type loci are created by three processes: the firstorder entry of individual free radicals into N . 1 type particles: the first order exit of single free radicals from N,,_, type particles: and the occurrence of mutual termination in N,,+e type particles. T h e same three processes occurring in N,,type particles lead to their disappearance: ,
[
t
,N,, l(P / N ) + N . + , ( k . , . s / N . 4 ) ( [ n + l ] / v S ) + N . , . . ( k , / N ~ )
([n+2][n+
= N,,{ ( p ' / N ) + (k,.s/N4)[,1Iv ~] 4 ( k , / N a ) [ n ( n   1)/v"] }.
l]/v") 127a)
This recursion formula may be simplified to
ceN. , + m N . + l ( n + 1 ) + N , , + . , ( n + 2 ) ( n + 11 = N , , { c ~ + m n + n ( n 
1)} (27b1
where c~ = (NAv*/k:) and m = (k:/k~). Significant progress in solving this recurrence relationship has been made by Stockmayer, c'2~ O ' T o o l e ~2a} and Gardon. c'4~Their contributions will be reviewed in turn. It eventuates that the general solution of the polymerization rate equation may be expressed approximately in terms of the limiting solutions obtained by Smith and Ewart. ~'~)T h e germinal, and in many ways definitive, studies of these authors will therefore be discussed first.
2.2,1,
The SmithEwart theory
Smith and Ewart ~I~)did not attempt to obtain a general analytic solution of the recursion formula because of its apparent mathematical intractability. Computers were not then readily available to provide a numerical solution. Consequently, Smith and Ewart were forced to rely upon explicit solutions derived for three limiting situations: (i) Case 1, when fi ~ 1. This requires, for example, that the rate of exit of the free radicals from the particles greatly exceeds their rate of entry (i.e. m >> c~). (ii) Case 2, which is characterized by h ~ 0.5. This occurs when the rate of termination is faster than the free radical entry rate and, in addition, transfer of free radicals out of the particles is negligibly small (i.e. m G ff < 1). (iii) Case 3, presumed by Smith and Ewart to occur when fi >> 1. This demands that the rate of entry of free radicals outpace their rate of
]58
A.E.
ALEXANDER AND D. H. NAPPER
termination (i.e. a > 1). It eventuates that case 3 kinetics holds approximately down to h  1.5. Cases 2 and 3 are the most important and these alone are discussed. 2.2.1.1. Case 2. T h e mathematical treatment of case 2 hinges on the simplification of the recursion formula (27b) to give an approximate expression for N,,_,/N,, in terms of a. This enables every term in the series for both n~ and N to be expressed as a product of No and a function of c~. No is cancelled out in forming the quotient n r / N = ft. The elimination of the terms involving N~+2 and N~+, from the recursion formula is obviously required. T h e N,,+, term is lost simply by setting ks (or m) = 0; this corresponds to zero probability of free radical transfer from the latex particle. T h e term involving N,,+2 is rendered unimportant by assuming, somewhat paradoxically, that kt is large (i.e. a is small) so that termination is very rapid. N,,+2 must therefore be negligible compared with N,, (or N,~_,). Thus the required relationship for n > 0 is Nn1/N,, = l
+ n ( n   1)/o~.
(28a)
F o r n = 0, the term in Nn+2 must be retained: N2 = Noc~/2.
(28b)
Also N3 = Noct~/2 (a + 6),
(28c)
N, = N 0 ( 4 a + 6 ) / ( a + 6 )
(28d)
T h e s e relationships yield ft = (1/2) ( l + a + 0 ( ~
2)
(29)
Since o~ is small, case 2 is characterized by ft  0.5 and lira ft = 05. Ot~0
Experimentally, this condition may be approached if the particles are sufficiently small or the initiation rate low, or both. For case 2, there is a decided kinetic advantage in having a large number of particulate compartments for the free radicals since the occurrence of mutual termination by free radicals in different particles is prevented. T h e wisdom of hindsight renders the result intuitively obvious: (2" each particle contains one free radical, which propagates until another free radical enters the particle when mutual termination occurs, almost instantaneously. This particle is dormant until another free radical enters, when propagation recommences. Each particle therefore polymerizes, on average, for half of interval 11. 2.2.1.2. Case 3. N o solution of the recursion formula is generated but the approximation is made that all particles contain the same number of
EMULSION POLYMERIZATION
159
free radicals 1i, where ~ >~ 1. The steady state condition is applied: p ' I N = 2 ( k d N A ) (h~lv '~)
(30)
where the factor of 2 allows for the annihilation of two free radicals in each mutual termination step. Clearly = ( p ' v ~ N ~ / 2 N k , ) 1~'2= (c~/2) 1/2
(31)
and (d[M]/dt)
= (k,,/NA)CMN(cx/2)'""
(32a)
= k~,C~l(p' Vi,f/2ktN~) 1/2.
(32b)
3'he rate of polymerization in case 3 is thus independent of the number of latex particles. N o kinetic advantage is won by having a large number of small particles. T h e rate depends upon the square root of the total volume of polymer. Accordingly a plot of the square root of the percentage conversion versus time is predicted Io be linear.
2.2.2.
The StoekmayerO'Toole theory
T h e mathematical restriction of rapid termination was forced on Smith and Ewart to enable them to obtain a solution of the recursion formula for one limiting case. T h e r e is no a priori reason why a restricted solution imposed by mathematical intractability should of necessity correspond to any physical reality. Indeed, as van der Hoff Ca~has observed, the rate of mutual diffusional termination of polymeric free radicals in latex particles of internal viscosity of the order of 103106 poise, must be relatively slow. This is confirmed by the recent calculations of Gardon, ¢~ to be discussed in Section 2.2.3. The intervention of initiator fragments, otigomeric free radicals or radicals formed by transfer reactions would seem necessary to permit rapid termination in even quite small particles. Obviously a solution of the recursion formula which removes this restriction on k~ is desirable. S t o c k m a y e r ¢'~2~has shown how a general analytic solution of the recurrence relation (27b) may be derived for all real, positive c~ and m. This proceeds through a generating function f ( ( ) . which when expanded as a power series in {:, contains the desired N,, values as real, positive coefficients: f ( ~ ) = Z N,d". /33) It=()
The value of ~ is obtained by differentiation: h = [df(sC)/d(se)]/f(()]~=~ = f ' ( 1 ) / f ( l ) .
(34)
160
A.E.
A L E X A N D E R A N D D. H. N A P P E R
Normalization of the distribution is unnecessary because h is a simple quotient. Successive differentiation of the generating function gives: 0c
f ' (~) = ~, n N ~ n',
(35a)
n=O
f"(£) = ~, (n) (n  1 ) N . £ ~2.
(35b)
n=0
It follows thatf(~:) satisfies the secondorder differential equation: (1 + ~ ) f " ( ~ ) + m f ' ( ~ )  a f ( ~ ) = 0.
(36)
This can be proved by supposing that f(~¢) satisfies the differential equation (36). Substitution of the infinite series (33), (35a) and (35b) should generate the recursion formula (27b).' On performing this substitution, the coefficients o f ~ n and ~n' yield ( n + 2 ) ( n + 1)N,~+2+m(n+ 1 ) N ~ + l  a N n + ( n ) ( n + 1)N~+, ~ 0,
(n) (n + 1)N,,+I + m n N .  aN,,_, + (n) (n  1) N .  O. Subtraction of these two identities yields the recursion formula as required. With the additional changes in variables p~ = 4a(1 +~:) andf(~:) = p l  m q ( p ) the differential equation (36) becomes
p2 (d2q/dp2) + p (dq/dp)  [ ( 1  m) 2 +p2]q = 0.
(37)
This is the standard form of a modified Bessel equation, (26~two particular solutions of which are
q = l+_(am)(P),
(38)
where Iv(p) denotes the modified Bessel function of the first kind of order v. The general solution is obtained by a linear combination of the particular solutions. Stockmayer, presumably on the criterion of physical acceptability discussed by O'Toole, (23~restricted the solution to
q = Iv(p) where v = [1m[. He thus calculated that  (a/4)(l_,,,(a)]ll_,,(a)), i f m ~< 1,
rl=([m1]/Z)+(a/4)(l,o_z(a)/Im_,(a)),ifm where a s = 83.
(39a) ~> 1
(39b)
16!
EMULSION POLYMERIZATION
T h e case of special interest when m = 0 gives h = (a/4)
(Io(a)/I, (a))
(40)
so lhat the rate of polymerization in this instance becomes  ( d [ M ] / d t ) = ( k v / N A ) C ~ z N ( ~ / 2 ) ' ~ 2 ( l o ( a ) / l , ( a ) ) .
(41)
Comparison (27) of eqn. (41) with the S m i t h  E w a r t case 3 rate eqn. (32a) shows that the influence of compartmentalization of the free radicals is expressed solely through the subdivision factor z * = [ l o ( a ) / l l ( a ) ] . A plot of z* as a function of a (Fig. 1) demonstrates that z* differs from unity by less than 10% if a /> 6, which corresponds to a relatively small limiting value of h ~> 15. Case 3 kinetics would therefore be expected to provide a reasonable description of systems for which h is greater than ca. 1"5. 10 N
o~
~SMITHEWART
"<
CASE 2
5
SMITHEWART D rn
CASE 3
. . . .
0
]0.5 0
11.0 [
12.0 5
~
13.0
~ 14.0 fi
10
15
I 20
(3
Fro. 1.
The subdivision f a c t o r z * as a f u n c t i o n o f the p a r a m e t e r a o r the average n u m b e r of free radicals per particle h.
O ' T o o l e (23) has criticized, on physical grounds, the particular solution adopted by S t o c k m a y e r when the rate of radical desorption is small but finite (i.e. 0 < m < 1). T h e general solution of the modified Bessel equation is simply a linear combination of the two particular solutions: ~c
f(~:) = ~ N,,~"
(33)
tl=O
= (1 +~)(1")/={A*l,_,,(a[(1 + ~ ) / 2 ] '~z) + B*]_(,_,,, (a [ (1 + ~')/2]'/2) }
(42)
where A* and B* are constants. If m is integral, the two particular solutions are identical (z6) and the S t o c k m a y e r formula (39b) is correct. Equally, if desorption is dominant (i.e. m > 1), the first particular solution diverges at sc =   1 , which implies the existeni:e of negative values of N,,. T h e latter is physically impossible..4 * must therefore be set as zero. Again the S t o c k m a y e r solution (39b) is correct for nonintegral m.
162
A.E.
A L E X A N D E R A N D D. H. NAPPER
O ' T o o l e has shown that to set B* = 0 for 0 < m < l, as S t o c k m a y e r has done, is incorrect. Differentiation of eqn. (42) and application of the relations b e t w e e n Bessel functions (26) gives N , , = (a2/8)"/Z(l/n!)[A*l~_,._.(a/X/2)+B*I.,+._l(a/X/2)].
(43)
W h e t h e r the A* term or the B* t e r m is physically acceptable may be determined by the a s y m p t o t i c b e h a v i o u r for large n: N. 
(a2/8) (J')/2 ( 1In !) { (A */~r) (n + m  2) ! sin (n + m  1 ) + B* (a2/8)m,+./(m  1 + n) !}
(44)
T h e first term is dominant in this limit. H o w e v e r , it alternates in sign and must therefore be rejected. T h u s B* # 0 for 0 < m < 1. T h e n o r m a l i z e d free radical distribution function, valid for all values of m, is thus
N . = f~ = a" 2("l3")121.,+._1(a/X/2)/n !1.,1 (a).
(45)
Nn = fn = (an/2 ('+zn)/z) 1,,1 (a/k/2)/n !11 (a).
(46)
Ifm = 0
Typical distribution curves predicted by this relation are presented in Fig. 2. T h e s e distribution functions m a y be used to calculate the instantaneous rates of polymerization, for given values of a and m, through eqn. (22). 1.0 a
~
0.1
._1
~
=
Z _o

o.ooi
FIG.
2.

_
I I I I I 0 2 ,~ 6 B 10 NUMBER OF RADICALS/PARTICLE (n)
The normalized free radical distribution functions computed using the O'Toole theory; © ~ = 1, • ~ = 2.03, • ~ = 4.01.
EMULSION
163
POLYMERIZATION
The instantaneous rate equations may then be integrated to yield the c o n v e r s i o n  t i m e curve. H o w e v e r , both computations are more easily performed through eqn. (26) and ~, expressed by O ' T o o l e for all m, as: =. (a/4) [l,,,(a)/l,,_i ( a ) ] .
(47)
A comparison of the ~ values of S t o c k m a y e r with those of O ' T o o l e is presented in Fig. 3. 1.0
m 0
0
I
2
~
/
4
/1
3
4
tl
FIG. 3. Variation of the average n u m b e r of free radicals ~ with the p a r a m e t e r a for different values of the radical e s c a p e p a r a m e t e r m : ( ) calculated using the O ' T o o l e theory, ( . . . . ) calculated using the S t o c k m a y e r theory.
For the particular case of m = 0, N a p p e r and Parts (''~) sought to integrate the instantaneous rate expression using the S t o c k m a y e r relationship (40):
d[M]ldt = (k~C~tNIN4)
(a/4) [lo(a)ll, (a)]
where a "~= ( 8 v p f N 4 / z k t ) . Moreover, [M] o  [M] = lO:3Nv,,d,,/M,,..
(41)
(48)
T h e s e relationships enable the rate of reaction to be expressed in terms of a:
 d [ M ] / d t = ( I O:~zl,tNd~,/4N4 M,,,f) a ( da/dt ) = (k,,CMN/N4) (a/4) [lo(a)/l~ ( a ) ] .
(49)
After integration
f~'i [1, (a)llo (a) ] da = (kvCMM,,,f/lO'~'ck,d,,)t.
(50)
164
A.E.
ALEXANDER AND D. H. NAPPER
The integration of the Bessel function ratio was not performed explicitly, but numerical integration showed that for a / > 2(h t> 0.72): f : [Ii(a)/lo(a) ]da ~ 0.93a 1.35.
(51)
Substitution of this approximation into eqn. (50) gives (% conversion) 1/~  (k~CM/0.93) (M~fN/8Oktdp [M] oNA~) 1/2t + 1.45(105ktdpNr]8f[M]oMwNA) 1/2.
(52)
This equation also predicts that, if ~ is sufficiently large for the approximation (51) to be valid, then the square root of the percentage conversion increases linearly with time. Equation (52) will be compared in Section 2.2.3 with a more rigorous version derived by Gardon. t24) To facilitate comparison, eqn. (52) will be expressed in terms of the volume concentration of formed polymer: Vp = I "42.4NPt~+ 0"78BSEt+ C NP
(53a)
where
ANP= O'102(kp2/ktNA)(dM/dp)2(C~M2/(1 (~M))P',
(53b)
B sE 0.5 (kp/Na) (dM/Np)C~MN,
(53c)
C uP = 26.3 (kt/Na)T( l  ~M)N.
(53d)
Ideal mixing of monomer with polymer has been assumed in this transformation; if mixing is not ideal, then dM and (1  6 M ) must be replaced by 1/~M and 1If respectively. C uP is often negligibly small (ca. 0.01) and may be safely omitted for most emulsion systems. 2.2.3. The Gardon theory Gardon <24)has noted that the foregoing theories of interval II make at least two approximations which might conceivably lead to significant errors. The first criticism relates to the application of the steady state approximation to reactive intermediates, in this case free radicals, whose concentrations may change appreciably during interval II. The second objection is that the solution of the recursion formula was obtained by assuming that vs is constant, whereas the value of vs must increase during the entire interval. The validity of these criticisms, of which perhaps the second is weightier, is discussed below. It is still not yet possible to obtain an analytic solution of the time dependence of N~, if both the assumptions criticized by Gardon are abolished. Gardon t26) has therefore generated an equivalent numerical
EMULSION POLYMERIZATION
165
solution by means of digital computations. T h e restriction of m = 0 is retained. Instead of the recursion formula, G a r d o n writes the more exact rate expression for n # 0: d N , , / d t = (p' / N ){ N , ,  1  N,} + (kt/ N AVs) × { N , + ~ 2 ( n + 2 ) ( n + 1) N , , n ( n   1)}.
(54)
If n = 0, the term in N,,_j is dropped. The volume concentration of polymer is a timedependent parameter and may be used to eliminate dt from the rate expression. The mass balance gives  d [ M ] / d t = 10a(d~/M,,,)(dV~,/dt)
(55)
whilst CM = IO:~dM6M/Mw, V8 = ( V p / N ( 1  6M))
(56) (57)
The formally correct expressions are obtained by replacing dM and (14~,vt) by (1/~M) and (l/f) respectively in all G a r d o n ' s formulae. F o r ideal mixing of m o n o m e r and polymer, both forms of the expressions are identical. Substitution of eqns. (55) and (56) into eqn. (26) yields d V p / d t = 2BSEt~
(58)
where B sE ( = 0.5 ( k ~ , / N a ) ( d M / d v ) ~ M N ) is the S m i t h  E w a r t case 2 rate, i.e. the polymerization volume rate which theoretically would be observed if ~ = 0.5. Relations (57) and (58) enable dt to be eliminated from eqn. (54): 2 ~ B S E r ( d N , , / d V ~ ) = { N,,_I  N,,} + ( l / a ) { N,,+,2(n + 2)(n + 1) N,,n(n  1)}.
(59)
F o r convenience, three dimensionless variables are introduced before numerical integration: z, a parameter proportional to the conversion: x, a parameter proportional to time; and ~P, a parameter whose order of magnitude is k t / k , . By definition z = O. 1297(V~,/'cBSE),
(60)
x = 0698(t/T),
(61 )
= ( k t / k ~ ) ( d ~ / d M ) ( 1  ~bw)/6M.
(62)
166
Clearly and
A.E.
ALEXANDER
AND
D. H . N A P P E R
dz/dVp = z/Vp
(63)
dz/dx = (dz/dVp)(dVp/dt)(dt/dx)  0.372 t~.
(64)
Combination of relations (60), (62) and (63), and division throughout by N, yields
h(df, ddz) = 3.81 {fnifn} + (q//z){f,,+2(n + 2)(n + 1) f,,(n)(n  1)}
(65)
where
~,f.= ~ (Nn/N)= 1 /'/=0
and
(66)
n=0 oo
t / = ~] n~,.
(67)
n=0
G a r d o n solved numerically relations (65), (66) and (67) using the mathematical boundary conditions that, at z = 0, f0 = f l = h = 0.5 and fn = 0 (n # 0,1). T h e s e calculations yield h as a function of z, for assumed values of • and z. T h e results obtained are compared with those predicted by the S t o c k m a y e r theory (for m = 0) in Fig. 4. F o r all practical purposes, the two solutions are equivalent. T h e explicit expressions of the S t o c k m a y e r  O ' T o o l e theory may therefore be used in the future with considerable confidence, without resort to numerical analysis. Values of h were predicted, with an accuracy of better than 8%, by the empirical relationship h = 0"5 (1 + [6"3 Z/aIF°'94])0"5 (68) which is equivalent numerically to the S t o c k m a y e r eqn. (40). The free radical distribution function derived by numerical analysis corresponds closely to a G r e e n w o o d  Y u l e <29>distribution function. This is a rather complicated distribution function analogous to the Poisson type, but somewhat narrower and with a variance unequal to the mean. It would appear preferable, however, to employ the simpler distribution function (46) derived by O ' T o o l e , to which the G r e e n w o o d  Y u l e function approximates closely. Both distributions are positively skewed (Fig. 2). This is expected intuitively because of the finite probability of the o c c u r r e n c e of particles containing more than twice the most probable number of free radicals. Once 1/had been calculated, G a r d o n derived the c o n v e r s i o n  t i m e curve by integration of eqn. (64). T h e boundary conditions, determined by the completion of interval I, were z = 03016 and x = 1.232. Typical
EMULSION
167
POLYMERIZATION
5F
3I
0 0
/
~ 2
4
L 6
8
10
~ 14
12
J 1 6 18
(2 FIG. 4. V a r i a t i o n o f the average number o f free radicals per particle n with the parameter a for m = 0: ( ) calculated using the Stockmayer theory, • calculated using the G a r d o n theory.
conversiontime curves obtained by Gardon are shown in Fig. 5. The point to note is that the curves are convex to the time (x) axis. In general, there is therefore strictly no rate independent of conversion and time in emulsion polymerization, if the basic postulates adopted by Gardon are correct. These curves fit, to better than 8% accuracy, the empirical correlation: z =
(O'05464/vd°"~4)x2+O'186x+O'0721(l

1' 14/~°~4).
(69)
Conversion from dimensionless to experimental variables shows that V v = A~;F + BSEt + C ~"
(70)
where A ~: = 0 . 1 0 2 ( k ~ / ' 9 4 / k t ° " 4 ) ( d ~ / d v ) ~ : ' 4 ( 1 / N 4 )[¢b~"~4/( 1   ~b ~/)°"4]p '
{ 71 )
C ~" = 0  5 5 6 ( 1   1 . 1 4 / W ° " 4 ) r B s~:.
(72)
168
A.E.
A L E X A N D E R A N D D. H. N A P P E R
G a r d o n erroneously gives the exponent of the (du/dp) term in A ° as 1.0, instead of the correct value of 1.94. This is not a trivial discrepancy since such an error may change the value of lip by ca. 30%. The C ° term is often practically of the order 0.01 litre polymer/1, of dispersion and may be safely disregarded in such systems. 15
I0 
~:I
~:Z.
~=I0
¥:Z.O
Z
~:I00 =
5
0
FIG. 5.
I
I
I
10
20 X
30
40
Theoretical c o n v e r s i o n  t i m e curves calculated by G a r d o n for interval 1t; z is the c o n v e r s i o n parameter, x is the time parameter.
Comparison of the empirical c o n v e r s i o n  t i m e eqn. (70) generated by G a r d o n ~24~with that (eqn. (53a)) derived on the basis of the S t o c k m a y e r eqn. (40), reveals obvious similarities. T h e B sE terms are identical. The only minor differences between A ° and A se are that the exponents 1.94 and 0.94 in A ° are replaced by 2 and 1 respectively i n A sP. T h e numerical coefficients preceding the ,4 and B terms also differ somewhat, so that less weight is placed on the B sE term and more on the ,4sP term in eqn. (53a). Presumably, this arises because in establishing his correlation, G a r d o n set the B sE term coefficient at unity so that eqn. (70) would be valid for all ~ i> 0.5. Equation (53a) is at best valid only for n / > 0.72. W h e t h e r it is possible to differentiate between the two eqns. (53a) and (70) experimentally is not yet known; however, the G a r d o n equation is the more general and ought to be the more accurate. T h e identity of the B sE terms, in eqns. (53a) and (70), and the SmithEwart case 2 rate has already been noted. T h e A t 2 terms are also closely related to the S m i t h  E w a r t case 3 expressions. Transformation of the integrated form of eqn. (32) into the current notation, assuming ideal mixing, gives for case 3: Vp
=
,4SEt2 = 0" 125(kp2/kt)(dM/dp)2( I/Na)[qSMZ/(1


qbM)]p't2.
(73)
EMULSION POLYMERIZATION
169
Here ASEt z is the S m i t h  E w a r t case 3 expression, which is independent of the number of latex particles. The formal parametric similarity between AsK:, A xe and A ~; is obvious. T h e equation derived from the S t o c k m a y e r theory b y N a p p e r and Parts may be rewritten as (~/> 072). Vp = l" 16A set" + 078BSEt + C w'.
(74)
T h e G a r d o n equation may be approximated in a closely analogous form, valid for all n: V~, ~ 082ASEt~ + BSEt + C". (75) If kt is large, the B sw term is dominant and the S m i t h  E w a r t case 2 results (i.e. ~ = ½). If kt is small, the A sE term predominates and the S m i t h  E w a r t case 3 rate is approximately obeyed (i.e. ~ ~> 1.5). For intermediate kt values, the G a r d o n equation approximates to the sum of the S m i t h  E w a r t case 2 and case 3 rates (i.e. 0.5 < ~ < i.5). One interesting interrelation between A sE and B sE exists. When kt is small, all the captured free radicals contribute to polymerization and the maximum rate of polymerization is 2A sv t This, by eqn. (58), must simply equal 2BSE(t/r). Thus AS~x = BSE/r. The close agreement between the predictions based on the S t o c k m a y e r O ' T o o l e theory and those of the more exact theory of G a r d o n is not purely fortuitous. G a r d o n rightly states that the former approach assumes that ~ is constant and, from this assumption, calculates the variation ot' with time. This logically inconsistent method is not confined to emulsion polymerization for it has a long and successful history of application in classical reaction kinetics. The steady state approximation was introduced into the theory of complex reaction mechanisms because analytic solutions of the simultaneous rate equations were prohibitively difficult. The criterion that must be satisfied to apply correctly the steady state approximation is not that the concentration of the reactive intermediate [R] should remain exactly constant. Rather, the requirement is that JR.] should be sufficiently small for (d[R.]/dt) to be considered infinitesimal when compared with the rates of change of the other reactants. A more complete discussion of the application of the steady state approximation is given, for example, by Frost and Pearson. c~°) ]'he quasistationary state approach is known to describe the course of the reaction in many complex systems with good precision. In an emulsion polymerization, [R.] is of the order of 10 s mole/l, of dispersion. Even if [R] increases by a factor of 102 during the period of polymerization (say, 104 sec), the average value of (d[R.]/dt) is only 1()~° mole 1.~ sec 1, whereas   ( d [ M ] / d t ) is of the order of 10 a mole 1.1 sec 1. Not surprisingly, the error introduced in setting ( d [ R . ] / d t ) = 0 is small, as confirmed by G a r d o n ' s exact calculations. m a x
"
170
A. E. ALEXANDER AND D. H. NAPPER
The rate of polymerization in interval I1 depends upon kv and h and thus upon the relative values of k~, and kt. S m i t h  E w a r t case 2 considers the consequences of large k~. G a r d o n ~2~) has developed a simple but ingenious method for calculating the minimum value of kt in latex particles. Because of the high internal viscosity of the particles, the free radicals may be assumed to be immobile in this calculation. A lattice model is assumed for the termination step; termination only occurs when free radical ends are positioned on adjacent lattice sites at the same instant. Mixing of m o n o m e r and polymer is assumed to be ideal. What is required is the relative probabilities that termination and propagation will occur in a latex particle containing n free radicals. Consider a free radical end on a lattice site about to undergo propagation with a single m o n o m e r molecule. N o n e of its contiguous sites can contain a free radical because termination would have occurred. After propagation, termination occurs if one of the sites adjacent to the new position contains a free radical. T h e number of such possible sites (tr) is simply the total number of adjacent sites in the new location, excluding the first site and any site adjacent to the latter. F o r a 3D hexagonal lattice, for example, o  = 8. T h e probability that any particular site is free radical occupied is (n  1)/L, where L is the total number of lattice sites ( = Vs/VM). T h e probability of termination occurring in the particle is ( o  n ( n  1 ) v J v O . If tM is the average interval between successive m o n o m e r additions, the rate of termination is (20"n(n1)VM/VJaINA). Thus 2o(n)(n  1)(VM/v*tMNA) = k~(n )(n  1)/V~ N A. (76) In the period tM, the volume of polymer generated by propagation is nv~t, if the volume change (usually a small decrease) which occurs on
polymerization is neglected. By eqn. (58)
Vm/tm = (kv/Na) (dM/dp) ~bM.
(77)
Combinations of (76) and (77) gives (kt/k,,) = 2o(dM/di,)dpM.
(78)
Relation (78) predicts the minimum value of (Xt/k~,) on the lattice model, since any diffusional termination must increase this ratio. Obviously ( k / k v ) m m is of the order 110, to be compared with experimental values which are of the same order or larger. T h e case of a limitingly small probability of termination may also be considered kinetically. As discussed above, this implies that A "m a x = BSE/r. Substitution of the relevant relations (53c) and (71) into the
EMULSION
POLYMERIZATION
17 I
0" 185 ch,~t(dM/d~,)/( 1  ~.~ ).
( 79)
equality yields
(kt/k,,)min
=
This differs from the expression given by G a r d o n c'~ only in the extra term (dM/dp) derived from a different A" value. Relation (79) implies even smaller values for (kt/k,,), in the range 0.011. The values of (kt/k~,)mi,, predicted and observed for latex particles are to be compared with the experimental values of 104[06 observed with typical bulk or solution polymerizations. The immobilization of the free radicals may thus reduce kt by a factor of 103104, for example. It is this reduction in the large value of kt for the bulk case, together with an increase in the volume of the particle, which prevents rapid mutual termination of the free radicals in the latex particle and permits a monotonic increase in ~ throughout interval 11. Actually, the S t o c k m a y e r  O ' T o o l e theory show's that increases, even if the latex particle volume remains effectively constant or, for that matter, decreases somewhat. The number average molecular weight ( ( M x ) ) of the polymer formed during interval I! is independent of the conversion, if SmithEwart case 2 assumptions apply, it may be calculated from ~'4~ (My) = Mw(rate of propagation/rate of termination)
= Mw (k,,C~tO.5 N / [ p ' / 2 N , ] )
(80)
= (2 × lO:3)dt,NaBSE/p '
= 103 kvdMd)M N/p'.
( 81 )
Expressions for (Mx) and for the instantaneous average molecular weight (Minst) in the general case of unrestricted h have been derived by Gardon. ~4~ Let A be the average number of terminated polymer molecules per particle formed in interval 1[. The number of free radicals absorbed from the aqueous phase to form, by mutual termination, this number of terminated polymer molecules is 2A. The number of free radicals absorbed per particle in time t is ( p ' / N ) t . The mass balance therefore gives
( p ' / N ) t =: 2A + ~.
(82)
Under practical conditions, h is usually negligible compared with 2,.\. Hence
A = 0.5(p'/N)t.
(83)
T h e average weight per particle is 10:3( V , / N ) d,,. Thus ( m x ) = 103 (V~,/N) N~d,,/A.
(84a)
172
A . E . ALEXANDER AND D. H. NAPPER
Elimination of A, using eqn. (83), gives (MN) = (2 × l03) (dpNA/p') (Vp/t)
(84b)
whilst elimination of t through eqn. (70), yields: (MN) : (4 × 103) (AGNAdp/BSEp')Vp/{ [1 + (4AG/(BSE)2V,)] 1/2 1}. (85)
The instantaneous average molecular weight is related to the integral average by M (MN) = (1/t)J0 Minstdt (86) which in this case gives Minst = (2 × 103) (dpNABSE/p ' ) ( l "q [4AG/(BSE) 2 ] Vp) 112.
(87)
If A G ~ 0, Minst reduces to the SmithEwart case 2 result (eqn. (81)) as required. If A a ~> B sE, Minst reduces to (103dpNAAG/p'), which is the expected value for SmithEwart case 3 kinetics. It is stressed that the foregoing molecular weight formulae all correspond to t = 0 at the commencement of interval II. The measured integral average molecular weight of the polymer present in interval II is always modified by the polymer formed during interval I. 2.3.
Interval HI
Interval III begins immediately the emulsion droplets vanish and ends when the conversion of monomer is complete. Monomer is present both in the particles and, to varying degrees, in the aqueous dispersion medium. Consequently, interval III may in general be divided into three subintervals. If the monomer is reasonably soluble in water, or in the polymer particles, the supply of monomer may be sufficient to permit propagation to continue as the rate controlling step for much of interval III. Ultimately, however, as monomer is consumed, the internal viscosity of the particle increases inordinately and the diffusion of monomer to the propagation sites becomes rate determining. A third subinterval may occur if a point is reached where the glass transition temperature of the monomerpolymer mixture just equals the polymerization temperature. The particles then transform from a highly viscous liquid state to a glassy state. Polymerization in the particles may virtually cease at this point because translational diffusion of the monomer and segmental diffusion of the polymeric chain ends occur only extremely slowly under such glassy conditions.¢31)
EMULSION POLYMERIZATION
173
The central problem in developing a theory of interval III resides in calculating the decrease in kt with increasing particle viscosity. This decrease is the wellknown geleffect, associated with the names of Norrish, ~3~)Schul2 TM and Trommsdorff.~34~The geleffect often manifests itself as an increase in the rate of polymerization in the latter stages of bulk polymerizations, when the termination reaction becomes diffusion controlled.~3" Some progress has been made in developing theories of the geleffect, although little application of these ideas to interval I I1 has yet occurred. For example, North ~z~~7~has suggested that the rate controlling step in the termination of the polymerization of alkyl methacrylates is not the translational diffusion of macroradicals, but rather the segmental diffusion of the radical chain end. A simplified model to account for differences in segmental diffusion has also been developed. ~3n~The application of the Kuhn and Kuhn theory ~38~for the diffusion of polymeric chain ends to the problem of termination of diffusion controlled polymerizations, has also been considered. ~37,39~ If the dependence of the rate of termination on the volume fraction of monomer in the latex particles were known then the course of the propagation controlled subinterval could be predicted by eqn. (26). Allowance must be made, however, for the variation of C , with time: this may be most readily accomplished by inserting into eqn. (26) the experimentally determined equilibrium distribution of monomer between the latex particles and the aqueous phase. Horie, Mita and Kambe ~31) have developed a theory for diffusioncontrolled propagation, which is applicable to bulk polymerizations beyond, say, 50% conversion. This theory, which is based upon a free volume approach ~4°~ to the calculation of the diffusion coefficients of monomer and polymeric radicals, should also be applicable to the diffusioncontrolled propagation regime in interval 111, before the glass transition temperature is reached. As yet, however, the theory contains several empirical approximations. Interval I I I deserves more attention than that which has been directed to it hitherto, because more than 50% of the monomer may be polymerized during this period.
3. EXPERIMENTAL 3.1.
Rates of Decomposition of Initiators
It is often assumed that the rates of decomposition of initiators in emulsion systems are identical with those observed in pure water. This assumption obviously need not always prove to be correct. It is known, for example, that the rate of disappearance of peroxydisulphate anions,
174
A.E.
A L E X A N D E R A N D D. H. NAPPER
which are frequently used to initiate emulsion polymerization, is increased greatly by the presence of certain organic compounds. ~41,42) Morris and Parts ~43)investigated whether various soaps and monomers, which commonly occur in emulsion systems, alter the rate of thermal decomposition of potassium peroxydisulphate at 50°C. Two anionic soaps (sodium ndodecyl sulphate and sodium nhexadecyl sulphate) and three monomers (vinyl acetate, methyl acrylate and acrylonitrile) were all found to hasten the disappearance of peroxydisulphate anions. Mixtures of monomers and surfactants also increased the rate. However, a fully fluorinated anionic surfactant (L1 159, ex 3M Company) did not change appreciably the decomposition rate, presumably because it is particularly resistant to oxidation. The rate of disappearance of peroxydisulphate anions was accelerated dramatically by the presence of vinyl acetate; onethird of the initiator had disappeared after only 1 hour. This observation agrees qualitatively with results reported by Patsiga. ~44)A smaller, but still significant, acceleration was induced by methyl acrylate and acrylonitrile. Both the anionic surfactants studied only increased the rate of disappearance of peroxydisulphate anions while unoxidized surfactant remained. Similar increases had been reported previously. ~4~) Various mechanisms for the chain decomposition of peroxydisulphate in the presence of oxidizable additives have been proposed, as discussed by Wilmarth and Haim. ~46)The significant point to note here, however, is that the assumption that the rate of production of free radicals in emulsion systems is identical with that in water, or is constant over the reaction times involved, could lead to serious errors. The use of oxidation resistant surfactants is also advocated in emulsion recipes to simplify the interpretation of experimental results. The experiments which have been considered hitherto have determined the influence of additives from changes in the rate of disappearance of peroxydisulphate anions. One uncertainty associated with such studies is that these changes may not truly mirror alterations in the rate of free radical production/47~a) If this does happen, the results obtained by such experiments could well be misleading when applied to emulsion polymerization. Data more relevant to emulsion systems would be obtained by the direct determination of the rate of generation of free radicals. ~4s) An alternative, but less direct approach, is to study whether surfactants alter the rate of homogeneous aqueous polymerization of a watersoluble monomer, such as acrylamide.~5°) Homogeneous rather than heterogeneous polymerizations must be studied because surfactants may influence the rate of polymerization in heterogeneous systems through effects on colloid stability/43) Specific interactions between the polymer, the monomer, and the initiator must also be avoided. Friend and Alexander tS°~ found that the rate of polymerization of
EMULSION POLYMERIZATION
175
acrylamide in water was not changed by the presence of sodium ndodecyl sulphate, whether at a concentration above ( 4 . 8 x 102M) or below (87 x 10 4 M) the c.m.c. (8 x 10 :~ M), when potassium peroxydisulphate was employed as initiator. Apparently the presence of this surfactant does not significantly alter the rate of free radical production, at least up to the maximum concentration examined. Renex 690, a nonionic alkylphenyl ether of poly(ethylene glycol), likewise did not alter the polymerization rate when added at a concentration (8 x 10:~ M) well above the c.m.c. (8 x 10 '~ M). Similarly, cationic surfactants below the c.m.c, caused no marked change in the rate of polymerization of acrylamide. Whether the overall rate of decomposition of the peroxydisulphate was increased by the presence of these surfactants was not determined. In marked contrast to the preceding results, Friend and Alexander found that cationic surfactants of the class nalkyl trimethyl ammonium bromide reduced the rate of polymerization of acrylamide when present at concentrations above the c.m.c. This rate reduction was shown to be more pronounced with increasing concentrations of soap in excess of the c.m.c. For equimolar concentrations of cationic surfactants, all above their respective c.m.c.s, the decrease in polymerization rate was most marked with the longer alkyl chain lengths IC,, < CH < C1~). H o w e v e r , equimicellar concentrations of dodecyl and tetradecyl trimethyl ammonium bromides produced comparable diminutions in rate. Concomitant with these rate reductions were parallel increases in the induction period before the onset of polymerization. The effects induced by cationic surfactants above the c.m.c, are most simply explained by postulating that some peroxydisulphate anions are specifically bound to the surfaces of the micelles, which possess a high positive surface charge density. The peroxydisulphate anions so bound decompose into free radicals at a slower rate than ions in free solution. This could result from the operation of an electrostatic cage effect. Strong specific interactions between micelles and their counterions are well known. In the present case, this strong interaction leads to the precipitation of a stoichiometric compound under suitable conditions. The above explanation cannot be invoked to explain the increased rate of decomposition observed for certain uncharged oilsoluble initiators, when present in emulsion systems. Cumene hydroperoxide, for example, decomposes in emulsion systems at a rate which is several times faster than that in styrene solution. ~'~1~ All the cumene hydroperoxide usually employed in emulsion systems is soluble but only a small fiaction (e.g. 1%) of the initiator is present in the water, the remainder being partitioned into the monomer droplets or latex particles. It has been postulated ~2; that the increased rate of decomposition in emulsion systems derives from the operation of what is essentially a reversed interracial
176
A . E . ALEXANDER AND D. H. NAPPER
cage effect. The decomposition of the hydroperoxide is presumed to occur at the particle/solution interface: the organic radical (~bC(CH3)20") enters the particle whilst the hydroxyl free radical (.OH) remains in the aqueous phase. This explanation is in agreement with the broad kinetic features of emulsion polymerization in these systems, which are characteristic of the entry of free radicals individually into the particles. The intrusion of interfacial phenomena even into the process of initiator decomposition underlines the intrinsic complexity of emulsion polymerization. 3.2.
Entry of Free Radicals into Particles and MiceHes
The factors which govern the entry of free radicals into either micelles or particles are still poorly understood. This uncertainty exists whether the free radicals are charged or uncharged. A detailed understanding of this problem is an essential Prerequisite to a proper knowledge of the processes involved in emulsion polymerization. Thermodynamically, primary free radicals, such as SO4" and .OH, would not be expected p e r s e to exhibit any marked tendency to partition from the polar water, where they are usually generated, into the considerably less polar environment inside a micelle or a polymer particle. Moreover, if the behaviour of SO4: can be taken as a guide, even the specific adsorption of such anions as SO4" at the particle/solution interface is probably quite weak, (52) unless the particle surface charge density is highly positive. <5°) A more realistic approach is to assume that the initiation of polymerization occurs in the aqueous phase by the addition of monomer molecules to the primary free radicals. Initiation in the aqueous phase seems probable even for very waterinsoluble monomers, such as vinyl stearate. <~3) This monomer addition generates oligomeric free radicals of the type " M p S O 4  . These oligomeric species are amphipathic molecules consisting of a polar head group and, if the degree of polymerization p is sufficiently large, a nominally insoluble oligomeric chain. Such oligomeric species might be expected to be incorporated more readily into micelles, or to adsorb more readily only to the surfaces of the latex particles, than the primary free radicals. The polar head groups, however, would be anticipated to remain at the particle/liquid interfaces, rather than be buried within the particles. An alternative to oligomer capture by the particles is the association of several oligomers, which may give rise to the formation of new particles. The existence of oligomeric free radicals has been inferred from a variety of experimental evidence. Several sparingly watersoluble monomers, for example vinyl acetate, methyl methacrylate, etc., may be poly
EMULSION POLYMERIZATION
177
merized in the complete absence of added surfactant, if potassium peroxydisulphate is used as the initiator. ('8'54'~5) The polymer appears in the form of a stable latex. Extensive dialysis does not seem to lead to desorption of the ionogenic groups, as is found, for example, with polystyrene latices stabilized by added surfactants. (56) This suggests that, when the surfactant is generated in s i t u , the potential determining sulphate halfester anions are firmly attached to, or even incorporated into, the particle surfaces. T h e s e dispersions may be deduced to be electrostatically stabilized because if polymerization is initiated by uncharged hydroxyl free radicals, stable latices do not result. (57) Both electrophoretic mobility measurements (Ss) and endgroup analyses (5"~)support this deduction. T h e presence of oligomeric surfactants may also decrease the surface tension of the aqueous phase in suitable circumstances. ('~5~) An upper limit to the average molecular weight of these oligomers for vinyl acetate aqueous polymerizations was determined by Priest. (54) A surfactantfree solution polymerization of vinyl acetate, initiated by peroxydisulphate anions, was quenched at the point of incipient turbidity. The average degree of polymerization of the formed polymer was ca. 50. This figure enables a lower limit to be set to the number of particles which must be present in order to suppress nucleation in aqueous polymerizations of vinyl acetate. (8) T h e assumption is made that the oligomer is soluble up to the critical degree of polymerization of 50. If it has not been incorporated into a particle by this point, the oligomer precipitates (a somewhat vague concept in this context) and participates in nucleation. If polymerization is not diffusion controlled, the interval between initiation and propagation is estimated (s) from a propagation constant of 2.8 x 106 cm3mole~sec 1 to be 0.07 sec. Adopting a reasonable mean value for the oligomer diffusion coefficient (1 x 10 ~ cm'~sec ~), the r.m.s, distance of oligomer movement is calculated from the Einstein equation to be ca. 4 × 104 cm. This value is comparable to the mean centretocentre distance, assuming a cubic lattice model, for a latex containing 2 × 10 TM particles/1. Thus at least this number concentration of particles is required to suppress nucleation totally. T h e precise value depends upon the particle size, the mechanism of imparting colloid stability, etc. Experimentally. a larger value (ca. 1 × 10 J5 particles/k) for the critical particle concentration lbr vinyl acetate polymerizations has been found necessary; (''8~this suggests that perhaps Priest's value of ca. 50 for the critical degree of polymerization is somewhat excessive because termination was not instantaneous. The assumed value for the propagation constant is also in doubt. Evidence has now appeared to show that the rate of entry of oligomeric free radicals into latex particles may be strongly influenced by the mechanism of imparting colloid stability to the latex particles, in general,
178
A. E. A L E X A N D E R A N D D. H. N A P P E R
colloidal particles may be stabilized in at least three different ways (Section 4): electrostatic stabilization, where stability results from charge interactions; steric stabilization, where stability is imparted by nonionic macromolecules; and electrostatic plus steric stabilization, which is a combination of the two preceding mechanisms. Methods have been devised for generating polymer latices, of comparable particle size, which are stabilized by one of these three mechanisms. ¢6°) These latices, at the same particle concentration, may be used as seeds for subsequent polymerizations in which no new particles are formed, as demonstrated by ultracentrifugation. Typical rate results (6°~ obtained for such seeded polymerizations of vinyl acetate are shown in Fig. 6, when potassium peroxydisulphate is used as the initiator. Strictly, the specified stabilization mechanisms refer to the initial poly(vinyl acetate) seed latices and so only the initial polymerization kinetics (say < 10% conversion) can be usefully discussed. The initial rate of polymerization was found to be fastest when the seed particles were stabilized solely by an electrostatic mechanism, in this case imparted by the adsorption of nhexadecyl sulphate potential determining anions. With the addition of a steric contribution (provided by poly(ethylene oxide) chains of viscosity average molecular weight 1.0 x 10 4) to the electrostatic stabilization, the rate decreased by a factor of c a . 4. However, the rate of polymerization observed for the seed latex stabilized entirely by steric effects was about double that observed for the case of electrostatic plus steric stabilization. This implies that the presence of the electrostatic stabilizing barrier halves the rate. lOO
50
S
50
100 TIME
FIG. 6.
150 (M]N.)
200
T h e rates of polymerization of vinyl acetate in seeded s y s t e m s which are stabilized differently.
179
EMULSION POLYMERIZATION
The simplest explanation for the foregoing results, which is in harmony with previous studies, is to attribute the different rates of polymerization to different rates of oligomer free radical capture by the latex particles. The bulk of the experimental evidence todate implies that the loci of polymerization in such seeded systems are the polymer particles swollen with monomer. (28.'~2)This view, however, is not universally accepted. ("~~;3) Nevertheless, it provides a compelling explanation for the characteristic sigmoidal shape of the rate curves, which may be attributed to an increase in the average number of free radicals per particle during polymerization. This increase derives from the restrictions to diffusional selftermination of the free radicals which are imposed by the highly viscous particle media. The rate of polymerization is thus determined inter alia by the rate of entry of oligomeric free radicals into the particles, where the environment is favourable for propagation. Any impediment to free radical enlry must therefore decrease the polymerization rate. ELECTROSTATIC
STERIC
ELECTROSTATIC ~ STERIC
+
~*+
FIG. 7.
.
+

~
+
Schematic representation of the three types of seed particles just prior to the commencement of polymerization.
Schematic representations of the three different types of seed particles just prior to the commencement of polymerization is given in Fig. 7. Colloid stability studies suggest that the poly(ethylene oxide) stabilizing moieties project into, and are dissolved by, the aqueous dispersion medium. (~4) Thus when a steric barrier is present, each seed particle is surrounded by a region of mean microscopic viscosity greater than that of water, probably by a factor of at least 20. (6'~) However, as the poly(vinyl acetate) core particles are swollen by monomer, probably only twothirds of the surface of each particle is initially covered. The rate of diffusion of the oligomeric free radicals through the viscous poly(ethylene oxide) chains is slower than through water, leading to a decreased rate of free radical entry. Polymerization proceeds at a correspondingly slower rate, in this instance at only onequarter of the rate. Moreover, transfer of free radical activity from the oligomers to the stabilizing moieties may
180
A.E.
A L E X A N D E R A N D D. H. NAPPER
also contribute to this deceleration. ¢6~ The relative importance of the microscopic viscosity effect and of chain transfer has not yet been established. The influence of the electrostatic stabilizing barrier may also be attributed to hindered diffusion. The Sulphate anion free radicals which give rise to most, if not all, of the oligomers are negatively charged, as are the electrostatically stabilized dispersions. The rate of incorporation of the oligomeric species into the seed particles is thus impeded by electrostatic repulsion. The combination of electrostatic and steric barriers is more effective than either of the barriers acting separately. Indeed, the electrostatic repulsion of the oligomers may be enhanced by the presence of the poly(ethylene oxide) chains, which reduce the dielectric constant of the dispersion medium in the immediate vicinity of the particle surface. Additional evidence that the presence of poly(ethylene oxide) chains impedes the entry of oligomeric species was derived from particle nucleation experiments. ~6°~As nucleation was absent from all of the polymerizations relevant to Fig. 6, sodium nhexadecyl sulphate was added to promote nucleation. The concentration of new particles formed when the poly(ethylene oxide) chains were absorbed on the electrostatically stabilized particles, was severalfold greater than that formed in their absence. Methyl acrylate in seeded polymerizations was found to give results which were qualitatively comparable to those obtained for vinyl acetate. ~6°~ Replacement, as the enthalpic stabilizers, of the poly(ethylene oxide) moieties by poly(vinyl alcohol) chains did not qualitatively alter the effect of the steric stabilizer in vinyl acetate polymerizations. Accordingly, the influence of electrostatic and steric stabilization barriers on the rate of polymerization in seeded systems may conceivably be a general phenomenon, important in many emulsion polymerizations. One notable exception would be the steady state polymerization of those systems for which h = ½. The importance of these results in discussing interval II kinetics is obvious (Section 2.2). The extent to which the degree of surface coverage, the molecular weight of the stabilizing moieties, etc., affects the role played by the steric barrier remains to be investigated. The general observation that the mechanism of imparting colloid stability may influence the kinetics is also important in fixing the major locus of polymerization in certain heterogeneous systems. If polymerization occurs predominantly in the aqueous phase then the rate of polymerization ought to be independent of the nature of the seed particles, to a first approximation. Conversely, polymerization probably proceeds in association with the particles if the stabilization of the particles influences the polymerization kinetics.
EMULSION POLYMERIZATION
181
The inferred influence exerted by the electrostatic stabilizing barrier on the rate of polymerization of vinyl acetate, initiated by charged free radicals, agrees with that deduced from a study of the emulsion polymerization of methyl methacrylate initiated by potassium peroxydisulphate. Zimmt
Number of Particles in Interval I
The SmithEwart theory predicts that the number of particles formed at tile end of interval 1 is proportional to the 06 power of the initial micellar
182
A. E. A L E X A N D E R A N D D. H. N A P P E R
surfactant concentration and to the 04 power of the initiator. Accordingly the rate of polymerization in interval II is expected to vary in an identical fashion. This prediction has often formed the basis of a test of the applicability of SmithEwart theory to emulsion polymerizations. The results obtained have often been in agreement with theory, within the limits of experimental error. (7a73) Such agreement, however, is at best a necessary condition to establish the validity of the SmithEwart theory; it falls seriously short of being a sufficient condition. The initiator and surfactant concentration exponents usually derive from the slopes of the plots, on a loglog basis, of rate versus initiator or surfactant concentration. These slopes are notoriously imprecise so that it becomes difficult to distinguish between an exponent of 0.5 and the values 0.4 and 0.6. Indeed, an exponent of 05 has sometimes been reported (72"74.75~ for the soap concentration dependence of the number of particles formed during interval I. The upshot of this inaccuracy is that exponents in general, and kinetic exponents in particular, are only weakly discriminating between various postulated models. Medvedev, (69~ for example, has surmised that emulsifier molecules, activated by interaction with initiator, adsorb on the surfaces of particles and micelles. Polymerization proceeds at the interfaces at a rate proportional to the product of the square roots of both the initiator and surfactant concentrations. Brodnyan(76~ pointed out that, on the basis of kinetic studies alone, no unequivocal differentiation between the two theories is possible. Kinetic experiments which purport to "prove" the SmithEwart theory also "prove" the Medvedev theory, and vice versa. Doubtless other theories also give comparable kinetic predictions. Measurements, other than those involving the determination of exponents, are therefore mandatory in any critical test of emulsion polymerization theories. Gardon ~77~ has considered the many studies which have determined the variation of the number of particles formed in interval  with the soap and the initiator concentrations. The situation appears to be very confused and no definitive conclusions can yet be drawn. Possibly half of the experimental studies on the emulsion polymerization of different monomers are in agreement with the SmithEwart theory; the remainder deviate considerably from the predicted 06 and 0.4 exponents. Admittedly some of the systems examined overtly violate the basic assumptions adopted in deriving the SmithEwart theory (e.g. acrylonitrile and vinyl chloride are usually nonswelling monomers). But even for seemingly wellbehaved monomers, such as styrene and methyl methacrylate, a concordance of results is still lacking. The micelles are ascribed a special role in the particle nucleation process in interval I of the Harkins scheme of emulsion polymerization.
EMULSION POLYMERIZATION
183
It is inside the micelles that nucleation is believed to occur by the entry of a (presumably oligomeric) free radical and the subsequent polymerization of solubilized monomer. The experimental evidence which supports this special role is as yet mainly circumstantial: for example, van der H o ~ (27) showed that the number of particles formed in the heterogeneous polymerization of styrene increased by a factor of c a . 10 when the concentration of surfactant is raised from below to above its c.m.c. Comparable accelerations of the rate of polymerization have also been reported. However, the polymerization of methyl acrylate, at concentrations below its saturation solubility displays no abrupt increase in the rate when the c.m.c, is exceeded. (7~) This suggests that the micelles may function primarily as surfactant reservoirs in such heterogeneous polymerizations and that perhaps they even play the same role in the emulsion polymerization of the more watersoluble monomers. (79) In principle, the surfactant reservoir hypothesis and the conventional Harkins theory may be differentiated experimentally. The Harkins theory predicts that the spatial volume elements inside the micelles, at the instant of particle nucleation, appear finally within the latex particles. The suffactant reservoir theory implies, in direct contrast, that these volume elements are ultimately located outside the latex particles. Todate no experimental evidence has been adduced to assess critically this difference in predicted behaviour. By suitably labelling the interior of the micelles, however, the ultimate fate of these domains ought to be determined unequivocably. 3.4.
Number of Particles in Interval II
The constancy of the particle number concentrations during interval I I is central to most theories of this region (Section 2.2). However, the determination of the absolute particle number concentration in any colloidal dispersion with particle diameters less than 5000 A is a formidable problem, if an accuracy of better than, say, 510% is desired. Few, if any, of the methods currently available (e.g. light scattering, ultracentrifugation, electron microscopy, etc.) meet this demanding requirement in any realistically critical appraisal. Not surprisingly, therefore, little conclusive evidence has yet appeared to prove that the particle number concentration actually remains constant during interval 11, even in the emulsion polymerization of styrene. Typical results obtained by van der Hol~ 2) for the particle number concentration during an emulsion polymerization of styrene are shown in Fig. 8. These results were derived from electron micrography. There seems little doubt that the number concentration of particles is constant beyond 50% conversion. This conversion, however, corresponds to
184
A. E. A L E X A N D E R A N D D. H, N A P P E R 2.0 

  v  
•
u
,~ 1.5
O//
•
x
1,0 i.
0.5 e,,
N ~r 7
o.o
i
I
20
i
I
1,0
l
I
L
I
60 80 CONVERSION (%)
I00
FIG. 8. Variation of the n u m b e r concentration of latex particles with conversion during the emulsion polymerization of styrene: • data of van der Hoff, (   I lines drawn by van der Hoff. ( . . . . ) alternative lines through experimental points.
interval 1II and not to interval I1. Below 50% conversion, the interpretation of the experimental results is equivocal, depending to some extent on the subjective predilections of the observer. Van der HofP 2~has considered the results to be evidential support of the thesis that the particle number concentration is constant in interval I I. An alternative, but equally valid, interpretation of the data is that the particle number concentration increases linearly with the conversion from 10% to 40% conversion. If correct, this implies that the number concentration of particles is not constant during interval I I. Part of the uncertainty in interpretation originates in the electron microscopic method used to determine the number concentration: the accuracy is poorer at lower conversions because of the reduced contrast on the electron micrographs of the smaller particles. Most determinations of the particle number concentration, which utilize electron microscopy, derive from the measurement of the appropriate mean particle diameter at a given conversion. One difficulty with this approach is that artefacts which alter the observed particle diameter may be unknowingly introduced during specimen preparation and examination. Another attendant problem is that any error in the determination of the particle diameter is magnified when the particle number concentration is calculated from the cube of the particle radius. Alexander and Robb (s°~ attempted to circumvent these uncertainties by developing a procedure which relies on direct counting of the number of particles and is independent of the determination of absolute particle
EMULSION POLYMERIZATION
185
size. Briefly, the method entails the spreading of a mixture of the latex and poly(vinyl alcohol) at the air/water interface of a film balance. The total volume of the latex spread per unit area of the film is thus known. The surface film is compressed to the required extent, and samples of the film removed and examined in the electron microscope. Direct counting of the number of particles in the grid apertures of known area enables the original particle number concentration to be calculated, Excess surfactant must be removed from the latex by dialysis before the poly(vinyl alcohol) film is spread. With this and other precautions, the particle number concentration may be determined with an accuracy of c a . +__ 10%. This method of spreading the latex particles has some advantages over previous procedures which utilized proteins as their spreading agents. (81.82) Robb (s3) applied this method to determine whether the number concentration of particles was constant during the period of uniform rate for the emulsion polymerization of styrene. The particle number concentration was found to increase, approximately linearly, from 1.0 × 1017 particles/litre of latex at 5% conversion, to 1.3 × 1017 particles/ litre of latex at 35% conversion. At this latter conversion, the particle concentration achieved its limiting maximum value. The accuracy of the method suggested that the observed increase was a real effect. Van der Hoff's results, as mentioned above, may also be interpreted as showing a comparable increase. Additional evidence to support this increase in the particle number concentration during the constant rate period was seemingly provided by the particle size distributions. A sample at the end of the constant rate period was strongly negatively skewed, with a long tail towards smaller particle sizes. This tail was apparently absent from the distribution at the beginning of the constant rate region, Whether the tail, which would consist of particles of diameter less than 200 A, was genuinely absent in this instance or simply unresolved by the technique employed, is not known because no lower limit to the particle size resolution achieved by the film b a l a n c e ' p r o c e d u r e was established. All distributions showed an apparent cutoff at c a . 200 A. This may, however, be a genuine effect, arising, for example, from the curvature of the electrostatic double layers associated with small particles; Is4) van der HoW :~) has also reported his conspicuous failure to prepare polystyrene latices with particles of average diameter smaller than 250 A. Certainly the appearance of a tail at higher conversions, if truly absent at lower conversions, would imply the occurrence of nucleation in interval II. H o w e v e r , the question is far from being resolved and further experiments with styrene, and other monomers, are required. The preceding observations, if correct, demand that particle nucleation can occur even in the absence of micellar soap. Undoubtedly, this is the
186
A.E. ALEXANDER AND D. H. NAPPER
case. The formation of stable, electrostatically stabilized latices by the polymerization of watersoluble monomers, such as vinyl acetate and methyl methacrylate, in the total absence of added soap, has been mentioned already. The oligomeric surfactants are generated in situ in such dispersions. Presumably it does not form micelles as it is both generated and consumed progressively as polymerization proceeds. The polymerization of sparingly watersoluble monomers, such as styrene, at concentrations of surfactant well below the c.m.c., is also known to generate dispersions. Robb, (83) for example, initiated the polymerization of styrene (4%), which had been emulsified by sodium ndecyl sulphate, with potassium peroxydisulphate. Two surfactant concentrations were studied, one above and the other below the c.m.c., as determined by a du Noiiy tensiometer in the presence of initiator but no monomer. The rate of polymerization was comparatively slow in both systems. Nevertheless, polymerization proceeded in the absence of micelles at a rate which was only half of that observed when perhaps 40% of the soap was in a micellar form. Similar experiments which demonstrate that nucleation can occur in the emulsion polymerization or copolymerization of styrene in the absence of micelles have been reported, for example, by van der Hofft27) and Staudinger.(85) These observations, however, do not prove that particle nucleation actually occurs during interval I 1. The presence of the latex particles may significantly inhibit nucleation. Seeded polymerizations in which nucleation is totally suppressed by the addition of preformed particles, are possible for most of the common monomers. Some ingenious experiments devised by Schulz and Romatowski, (86"8~)which allow in part for the presence of the latex particles, are also of direct relevance to the question of nucleation in interval II. An emulsion polymerization of styrene was allowed to proceed to 11.5% conversion. For this polymerization, the monomer was initially present at ca. 15% v/v concentration, sodium ndodecyl sulphate was employed as the emulsifier and initiation was effected by the photochemically induced decomposition of a,~'azoisobutyramide. A sample of the emulsion was removed at 11.5% conversion and diluted about sixfold to reduce the surfactant concentrations to well below its c.m.c. value. The styrene and initiator concentrations in the diluted systems were then increased to their original values in the unpolymerized emulsion recipe. When photoinitiated polymerization was resumed, it proceeded at a rate which was only marginally less (ca. 30%) than that in the undiluted system. If no nucleation had occurred after the resumption of polymerization in the dilute sample, then a reduction in the rate by the dilution factor of about 6 would be expected. Romatowski and Schulz (87)attributed the failure of this prediction to the occurrence of nucleation in seeded
EMULSION POLYMERIZATION
187
systems below the surfactant c.m.c. Electron micrographs of the latices taken after the resumption of polymerization confirmed the expected bimodal distribution of particle sizes. It was inferred from these observations that new particles could be generated in the aqueous phase in the absence of micelles. Another interesting feature of these results is that the polymerization rate was found to be constant, even up to quite high conversions, when new particles were apparently being formed. Robb ~s3~ also observed a constant rate of emulsion polymerization of styrene when the number of particles seemingly increased. Romatowski and Schulz
Interval II Kinetics
G a r d o n <'4) has proposed theoretically that in interval 11, the rate of polymerization takes the form of eqn. (70): Vv ~ A<~t"2+ BS~t.
(70)
Equation (70) might be viewed as an arbitrary series expansion in t which may be fitted, more or less, to all experimental results. H o w e v e r , the coefficients A <~and B sE are not arbitrary because, to a first approximation, eqn. (70) is a linear combination of S m i t h  E w a r t case 2 and case 3 rates. A crucial test of the validity of eqn. (70) is therefore whether the values of A <: and B s~ derived from experiment agree with those calculated theoretically. Equation (70) implies that the conversiontime curve is not in general linear in interval I I, because the number of free radicals per particle is not constant during interval I I. Approximate linearity may be achieved, however, if BSEt ~ A~;t 2. G a r d o n ~8) has listed some of the experimental investigations which report c o n v e r s i o n  t i m e curves convex to the time axis, as demanded by eqn. (70). These include the emulsion polymerization of styrene, butadiene, isoprene, methyl methacrylate, methyl acrylate and vinyl acetate, lfAC;t2 >>> BS~:t, then a plot of the square root of the conversion versus time ought to be linear in interval 11, as has been observed for butadiene, methyl methacrylate and vinyl acetate) ss) This does not hold in general, however, as both terms of eqn. (70) must be included.
188
A.E.
ALEXANDER AND D. H. NAPPER
Values of both A c and B SE can be derived by a leastsquares fit of eqn. (70) to the experimental curves. Unfortunately, A ~ (eqn. (71)) is not very often amenable to theoretical calculation because the ratio of (kt/kp) is usually unknown for the conditions prevailing within latex particles. The best approach at present is to determine whether the values of (kt/kp) derived from the experimental A c values are of a reasonable order of magnitude. Values of (kt/kp) in the range 1103 were obtained in this way for different monomers by Gardon) 88) These values are considerably smaller than the corresponding figures for bulk or solution polymerization but are considered by Gardon to be theoretically reasonable (Section 2.2.3). The B sE values obtained experimentally were in fair agreement with those calculated theoretically; the difficulty is that considerable latitude is possible in the selection of the literature values of the various parameters which are used in such calculations. 3.6.
Mechanism of Termination
It is usual to assume in discussions of emulsion polymerization that the probability of free radical escape from the latex particles is zero. This assumption is obviously incorrect if transfer is absent, because it implies that n can never tend to zero. Indeed, the usual SmithEwart case 2 and case 3 assumptions do not admit of n values less than 0.5. Of course, the StockmayerO'Toole theory (Section 2.2.2) is able to include the possibility of free radical escape. The importance of the possibility of radical escape in certain emulsion systems has perhaps been overlooked in the past. Schulz and Romatowski <86) investigated a proposal by Bianchi, Price and Zimm
EMULSION POLYMERIZATION
189
rates and radical concentrations in these systems implied that the apparent halflives of the propagating radicals ranged from below 1 minute for vinyl acetate to tens of minutes in the case of styrene. The acrylic monomers possess intermediate halflives. The emigration probability of low molecular weight radicals for styrene was found to be ca. 5  1 0 % , which is considerably smaller than the value reported by Schulz and Romatowski. (86) Nevertheless, a mechanism is provided by which ~ can quite rapidly decay to values of less than 0.5 and, in the limit, can approach zero if initiation ceases. The probability of free radical escape from the latex particles would appear to be particularly high for monomers, such as vinyl acetate, which transfer free radical activity readily to the monomer and which are reasonably watersoluble.
3.7.
Poorly Watersoluble Monomers
Where deviations from the conventional S m i t h  E w a r t behaviour are observed, these are commonly attributed to the different (usually greater) water solubility of the monomer. Often the contribution of a solution polymerization c o m p o n e n t is invoked. (6163) Vinyl stearate is interesting in this context because it represents an extreme case of a monomer with a very low saturation solubility in water (7 × 10 7 M(9~) compared with 37 × 10:~ M for styrene(~3)). Polymerization in the aqueous phase should therefore be negligible. As poly(vinyl stearate) is soluble in liquid vinyl stearate, S m i t h  E w a r t type kinetics might be expected to prevail in the emulsion polymerization of vinyl stearate. T w o provisoes to this prediction should be noted: first, free radical transfer to vinyl stearate monomer is two orders of magnitude greater than the corresponding rate for styrene: (94) second, the Harkins mechanism of emulsion polymerization demands that the rate of mass transfer of monomer from the emulsion droplets through the aqueous phase to the reaction loci in the latex particles should be sufficiently rapid not to be rate determining. This requires that the monomer be sufficiently soluble in water. Precisely what is the lower solubility limit, and what parameters determine this limit, is not yet known. Flory (9~)has suggested a figure of 10 '~ M. Moore (~3) studied the emulsion polymerization of vinyl stearate, using potassium peroxydisulphate as the initiator and an isooctylphenol adduct of poly(ethylene oxide) as the nonionic stabilizer. T h e time dependence of the instantaneous rate was qualitatively similar to that observed for styrene, except that no T r o m m s d o r f f effect was evident in the latter stages of polymerization. A comparison of the maximum rates of polymerization (effectively interval II kinetics) of vinyl stearate and styrene is lhus permissible. T h e results obtained are sufficiently unusual to warrant special consideration here.
190
A. E. A L E X A N D E R A N D D. H. N A P P E R
Many specific features of the emulsion polymerization of vinyl stearate diverge from those observed for styrene. The number of latex particles generated was found to be independent of the initiator concentration, while the rate of polymerization was almost directly proportional to it. The maximum rate of polymerization did not vary as the 0.6 power of the surfactant concentration, as often observed for styrene. (2"3~ Rather, a complicated variation with emulsifier concentration was observed: when the monomer was almost completely solubilized in the suffactant micelles (0.366 M), the polymerization rate was very slow; however, as the number of micelles was reduced and the number of emulsion droplets increased, the maximum rate itself increased to an extremum at c a . 0.020.05 M surfactant. Further reduction in the surfactant concentration decreased the rate. The final number of latex particles formed was considerably less than the original number of micelles. The relatively small latex particle size implied the absence of significant polymerization in the emulsion droplets. The polymer was of a relatively small integral average degree of polymerization (e.g. 20, corresponding to an integral average molecular weight of c a . 6 × 103); this presumably is a consequence of the high probability of free radical transfer to monomer, each free radical being found to generate several polymer chains. Calculations based on the saturation solubility of vinyl stearate suggest that, even for such a sparingly soluble monomer, initiation by the primary free radicals may well occur in the aqueous solution. Free emulsifier stabilizes the polymer radicals so formed. Mass transfer of the monomer from the emulsion droplets to the free radicals was postulated to proceed via two different mechanisms: by diffusion through the aqueous dispersion medium; and by "sticky" collisions (i.e. collisions whose lifetime is considerably longer than elastic collisions) between the latex particles and both the monomer droplets and any micelles which remain. Collisions between monomer droplets and latex particles might be expected to be particularly fruitful for this type of monomer exchange. The direct proportionality of the rate of polymerization to the initiator concentration was considered to be a consequence of the direct dependence of the number of free radicals generated per unit time on the initiator concentration. Given time, however, the systems of lower initiation rates catch up in free radical production with the others; hence the final particle number concentration is independent of initiator concentration. The curious dependence of the rate of polymerization on the surfactant concentration was explained by supposing that when all the monomer is solubilized in the micelles, polymerization occurs in the micelles by absorption of oligomeric free radicals. Mass transfer of monomer to the growing particles is necessary to permit further propagation. However, collisions between micelles and latex particles are relatively inefficient
EMULSION POLYMERIZATION
191
for monomer transfer and so polymerization proceeds slowly. On reducing the surfactant concentration, less monomer is solubilized in the micelles and more monomer appears in the initial reaction charge as emulsified droplets. These droplets are more effective than micelles in yielding monomer transfer to latex particles during collisions; hence the rate of polymerization increases as the surfactant concentration decreases. Ultimately, however, the reduction in the surfactant concentration so decreases the micelle concentration, and hence the number of particles formed, that high rates of polymerization are no longer attained. The pattern of behaviour which emerged for the emulsion polymerization of vinyl stearate is somewhat reminiscent of that found by Gerrens and Kohnlein(95) for 2,4dimethylstyrene. This monomer would be expected to exhibit a watersolubility intermediate between that of vinyl stearate and that of styrene. The number of latex particles formed in the emulsion polymerization of 2,4dimethylstyrene was found to be relatively insensitive to the initiator concentration but directly proportional to the surfactant concentration. The broad polymerization features were intermediate between those of styrene and vinyl stearate. The results discussed in this section highlight the hazards which are involved in attributing any special behaviour observed for the emulsion polymerization of styrene to the waterinsolubility of that monomer. Such behaviour could be ascribed, with equal validity, to the high watersolubility of styrene, relative to vinyl stearate or 2,4dimethylstyrene. 4.
COLLOID
STABILITY
The stabilization of latex particles against flocculation is clearly important in emulsion polymerization. In fact, this is partly the raison d'etre for the presence of surfactants in emulsion polymerization recipes. Little attention in the past has been paid to the role of colloid stability in such polymerizations, apart from consideration of the pragmatic question of whether stable latices are formed or not. This attitude is now seen to be untenable in general. Some of the more recently developed concepts of the mechanisms whereby colloid stability may be imparted will therefore be considered briefly. Dispersions of uncharged, uncoated latex particles flocculate rapidly as a consequence of the longrange London dispersion forces operative between the particles. (97) The potential energy of attraction (VA) between two spheres of radius r , whose centres are separated by a distance R, is given by the microscopic, dipole theory as
1/4 ~   A ' { 2 / ( s * 2   4 ) + (2/s'2)+ In ([s*"4]/s*2)}/6
(88)
where s * = (R/r) and A ' = effective Hamaker constant. ¢~4) When the
192
A.E.
A L E X A N D E R A N D D. H. NAPPER
distance of closest approach (Ho) is small compared with the particle radius, eqn. (88) simplifies to VA   A 'r/12H0.
(89)
T h e effective H a m a k e r constant can be calculated from the vacuum H a m a k e r constant; allowance must be made for relativistic retardation effects, (98) for the dielectric constant of the dispersion medium, (99) for the presence of any adsorbed species, tl°°) and for an Archimedean effect of the dispersion medium, t98) V a c u u m H a m a k e r constants are calculable from suitable measurements of optical refractive dispersion, (1°1) ionization potentials, (1°2) interfacial tension, (1°3) diamagnetic susceptibility (1°4) or optical data relevant to the Lifshitz theory. (~°~) Calculated values o f A ' for latex particles dispersed in water usually fall in the range ( 1 10) × 1013 erg. Consequently dispersion forces are operative over tens of angstroms.
+
V'r
\
0
R~iU .ELECTROSTATIC LSION~ STERIc
_
vA
/ !
/
/
Ho
LONDONATTRACTION
FIG. 9. Potential energy diagram illustrating the interaction between two stable colloidal particles as a function of the distance of separation H0; (. . . . ) attractive and repulsive potential energies, ( ) total potential energy Vr.
T h e attractive potential energy of two latex particles as a function of their distance of separation is shown in Fig. 9. Also shown in this diagram is the shape of the total potential energy (Vr) curve which is necessary for stability. T h e characteristic feature which presages stability is t h e presence of the potential energy maximum (VM). This functions as an activation energy, which, if sufficiently large (i.e. VM >> kT), prevents frequent irreversible entry of the particles into the primary flocculation minimum. In this way, a longterm kinetic type of stability is generated.
EMULSION
POLYMERIZATION
193
']'he potential energy maximum is normally produced by incorporating a repulsive potential energy between the particles. This may be achieved in three different ways: electrostatically; sterically; and by combination of both electrostatic and steric effects. ']'he electrostatic charge associated with latex particles usually originates from the adsorption of potential determining surfactant ions. or from the presence of ionogenic groups at or near the particle surfaces. The ionogenic groups, e.g. carboxylic acid or sulphate halfester groups, may originate in the products of the breakdown of the initiator or from adventitious oxidation of double bonds. The DLVO theory, due to Deryagin and Landau (~°6) and Verwey and Overbeek, ¢1°') is considered to be a definitive theory of electrostatic stabilization. It shows that to a first approximation, the repulsive potential energy (Vn) between two aqueous latex particles, close to incipient flocculation, is given by
VR =
(~r~2/2)
exp (KH0)
(90)
where ~ = zetapotential of the particles and the Debye double layer thickness l/K, expressed in nanometres, is given by (0.3/11/2), 1 being the ionic strength. The total potential energy (VT) is simply the sum of VH and VA. The DLVO theory explains in a qualitative fashion at least, many of the properties of electrostatically stabilized dispersions. It is less successful quantitatively.¢1°s)But it explains, for example, why small latex particles are in general less stable than large particles, as a result of the curvature of the double layers associated with small particles. This is important in considering the genesis of particles in certain latex systems. ('s,79) Whether the capture of charged oligomers by charged particles or micelles is amenable to a DLVO type of approach remains to be decided. The second general method of stabilizing colloidal particles utilizes nonionic polymer chains. No definitive theory comparable to the DLVO theory, has yet been propounded for such steric stabilization. Certain general principles have emerged, however. The repulsive potential energy in steric stabilization originates in the interpenetration and compression of the stabilizing polymer chains during Brownian collisions. "°~) The associated Gibbs free energy change must be suitably positive for stability. This may be achieved in three different ways, depending upon whether enthalpic, entropic or a combination of both effects provide the dominant positive contribution. (la°) Enthalpic stabilization is most common in aqueous systems: for example, lattices stabilized by poly(ethylene oxide) or poly(vinyl alcohol) adducts are enthalpically stabilized. Enthalpic stabilization is usually characterized
194
A.E.
A L E X A N D E R A N D D. H. NAPPER
by flocculation on heating; this contrasts with the normal flocculation on cooling of entropically stabilized dispersions. The point of incipient flocculation of sterically stabilized dispersions can often be correlated with the point at which the dispersion medium becomes a theta (O)solvent for the stabilizing moieties. ¢1°a1°~ This is in agreement with the predictions of the FloryHuggins theory for the processes of polymer interpenetration and compression. Thetasolvents are characterized by zero nett Gibbs free energy change for both those processes and this corresponds effectively to the removal of the repulsive potential energy barrier. The requirement that the dispersion medium must be a better solvent for the stabilizing chains than a Osolvent places an immediate restriction on the nature of the dispersion media which can usefully be employed in emulsion polymerizations utilizing steric stabilization. It determines, for example, the ionic strength which can be tolerated at a given temperature for aqueous emulsion polymerizations using poly(ethylene oxide) or poly(vinyl alcohol) as stabilizing chains. This ionic strength, at not too elevated temperatures, is several orders of magnitude greater than that tolerated by electrostatically stabilized dispersions; at higher temperatures, however, it decreases dramatically.
5.
CONCLUSIONS
The current status of the theory of emulsion polymerization depends critically upon whether or not the Harkins scheme is an accurate description of the actual processes by which such polymerizations proceed. If it is a reasonably true picture, then the kinetics of interval II are in principle known to any desired accuracy from either the explicit theory of Stockmayer and O'Toole, or the more elaborate computational procedures devised by Gardon. In practice, it seems likely that the StockmayerO'Toole theory is of sufficient accuracy to permit meaningful comparisons of theory and experiment. Interval I remains a theoretical stumbling block. The assertion by Gardon that the SmithEwart lower limit calculation predicts the actual number of particles generated in interval I, is unconvincing as it relies upon a grossly oversimplified physical model. More direct studies of the role of the micelles in particle nucleation appear warranted. Some of the difficulties associated with the transition from interval I to interval I1, inherent in the SmithEwart theory, may be removed by postulating the preferential entry of free radicals into the latex particles. Precisely what factors govern the entry of free radicals into particles and miceiles, however, remain to be elucidated.
EMULSION POLYMERIZATION
195
The theory of interval III, during which perhaps 50% of the monomer is polymerized, is still very rudimentary. Experimentally, it would appear that orthodox kinetic experiments are unlikely to establish unambiguously the correctness, or otherwise, of the Harkins model of emulsion polymerization. More emphasis should therefore be placed on locating (by physical methods) the whereabouts and ultimate fate of the polymerization loci. The importance of the mechanism of imparting colloid stability is now recognized. Some doubt has been raised as to whether the number of particles remains constant during interval I1 of the emulsion polymerization of styrene. The occurrence of nucleation during this interval, under certain circumstances, has also been demonstrated. The quite significant probability of free radical escape from latex particles indicated for most common monomers has been attributed to chain transfer to mobile monomeric species. Monomers less watersoluble than styrene do not apparently obey SmithEwart kinetics. As the same is often said of monomers more watersoluble than styrene, it appears that styrene is somewhat atypical in its emulsion polymerization behaviour. Certainly the process of emulsion polymerization is more complicated and rather more subtle than was at first envisaged. 6.
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