conversion prediction in high conversion free-radical poiymerizations

8/13/2019 Conversion Prediction in High Conversion Free-Radical PoIymerizations

1/11

Conversion Prediction i n High Conversion Free-RadicalPo Iyme r za ions*

S. BALKE, L. GARCIA-RUBIO, and R . PATEL

Xerox Research Cent re of Ca n a d aMississauga, Ontar io L5K 2L1

The objective of this articlc. is to introduce a new general kip-proach for conversion prediction i n high-coitversion free-radical polyiirerizatioiis-'"l'he hleth od of Kinetic Similarity."Using a sirigle homopolymer reference curve of methyl meth-acrylate XIRIA) and one to two time-scaling parameters, th econversioit cs . tiiiie curves for AlhlA, ethyl methacrylateEA IA ) ,vinyl chloride (VC), ieq lollitrile ( A N ) , styrene meth) 1

methacrylate S h l h l A ) , and st>-rene icrylonitrilr S A N ) couldbe well-descri1)etl. For h l ;\IA and VC, these parameters areshown to o b e y expectetl Arrhenius relationships over 45 to90C a n d 30 to 'iO C, respectively. The method is simple to kip-pl ., con tin rio us o v e r the whol e conversion range, utilizes e x -istiiig knowledge at low a i i d liniiting conversions, a n d avoidsapp1ic;ition of i i s i i l low coltversion ;issumptions into thediffiisioir-c.oiitrolle~1 egion.

IN T R O D U C T IO N

casurement of conversion of monomer t o polymerM uri ng a polymerization provides accurate, pre-cise data of great utility in process simulation a nd

control (1-3). This usefulness originates from severalaspects

Conversion is an important product property;since residual monomer levels are part of a reactorpro duct specification, reactor system desi gn an dcontrol mu st take conversion as a function of timeinto account.Conversion is a prime determinant of t ransportpheno mena within the reactor. I t translates into adecreased free volume and increased viscositydu r ing the reaction. As a result , reactions becomediffusion-controlled, and mixing difficulties, as

well as heat transfer problems, increase.The rate of increase in conversion is n ee de d if rateof heat generation within the reactor is to be es-t imated .Conversion us . time provide s a link to other poly-mer propert ies (3). For example, molecularweight distribution, copo lyme r composition dis-tribution, a nd seq uence length distribution canbe considere d as the sum of instantaneous distri-butions, each weight ed by a conversion incr ement(4, 5 ) .

Despite its importance, and although recent researchutilizing free volume (4, -10) and polymer entangle-ment theories (11-15) have provided valuable insightinto th e mechanism of diffusion control, pr ediction ofconversion v s . tim e is far from satisfactory from an engi-neering viewpoint.

For maximum engineeri ng usefulncss, criteria thatany conversion prediction mod el sho uld satisfy incl udethe following:

General applicability to different monomers,

homopolymerizaton, a nd copolymerization tocomplete converson.Ready incorporation of existing knowledge of aparticular polymerization.Absence of the stationary state assumption be-yond low conversion.Continui ty an d simplicity: one equation to d e-scribe t he whole conversion rangc without seg-mentation a nd without numerical solution dif-ficulties.N o necessity for molec ular weight information inthe model .Minimal reliance on literaturc values of rateconstantsAllowance for kinetic mechanism changes beyon dlow conversion.

From t he vast n umb er of publications in this area, theinformation most relevant to t hese criteria are:

Kinetic model s for low conversions ar e generallyavailable a nd satisfactory.Some very useful models to high conversion forspecific homopolymers are available e.g. , 4, 7 ,10, 16-19). For copolymers, much less is known(5, 20-23).Fina l (limiting) conversion can ofte n

bepredicted

by assuming that the reaction ceases when theglass transition temperature (Tg) of the reactionmixture equals the polymerization temperature.Fre e volume theory has bee n used for the estima-tion (4-8, 9).

POLYMER ENGINEERING AND SCIENCE, AUGUS T,1982, Vol. 2 2 , No. 12 777


2/11

S. Bulke, L. Garcia, u n d R . Putel

Correlations of rate parameters with free volumehave so far provided the most practical microen-vironment description for correlation of rateparamters 4, ). The correlation shown as Fig. 19in Ref. 4 f a normalized group of rate parametersus . free volume successfully superimposed datafor methyl met hacrylate polymerization at 70 and90C each at two levels of free radical initiatorover wide conversion ranges. Later, using thesame data, the individual termination an d prop-agation constants were separately calculated andshown to obey a similar plot, this t ime for 50, 70,and 90C ( F i g s . 2 and 3 in Ref. 7). Although cer-tainly not totally satisfactory, particular ly with re-gard to accuracy at all conversions, this enablesrapid, direct estimation of these rate parametersby calculation of free volum e from a specified re-action temperature and conversion.

Molecular weight effects on conversion may bepresent via entanglem ent of terminating macro-radicals (i.e., molecular weight information maybe a necessity in any conversion mod el) (9, 14, 15).Very recent theoretical developments utilizingthe reptation model of de Genne s for polymer dif-fusion (24) or combining entanglement theorywith free-volume theory (25) re-enforce thislikelihood.

Primary radical termination and transfer to poly-mer represent probable mechanism additions athigher conversions (13, 26, 30). Polymerimono-mer solubility effects (17, 19), an d violation of thesteady-state assumption (26) can also readily in-troduce complications.

This objective of this paper is to initiate the develop-men t of a new approach that can satisfy most a nd , per -haps for many polymers , eve n all of the above criteria.Two underlying principles are involved:

Formulation of the rate equation with the addi-tion of an unknown general time function rcprc-senting nonidealities in the kinetics. This is incontrast to the usual assumption of a specific

function of conversion to repres ent a group of rateconstants.

Definition of this unknown function by appeal tosimilarit y with a single known conversion 6s. timecurve. Often, this shape is implicitly considered informulation of a mathematical model for a poly-merization. However, explicit use of similarity ismuch less frequent. A singular example is the em -pirical interpolation between component homo-polymerization curves of styrene and acrylo nitrilein orde r to estimate the curves of the intermediatecopolymers for purposes of experimental design

( 2 3 ) . In other fields, examples are more readilyfound. In statistics, families of sigmoidal curveshave been generated by scaling and shifting ofaxes (28). In polymer rheology, Time Tempera-ture Superposition (29) has been a very useful re-lated concept.

THEORY

In the development that follows, homo-polymerization is considered first, followed by copo-lymerization. In ea ch case, the rate equ ation is first for-mulated, and then kinetic similarity is defined.

Homopolymerization

Formulation of th e Rat e Equati on. At low conversionand using the steady state assumption:

where

and

v = V,, 1 E X l b )x = conversiont = timec,, = initial initiator concentrationk , = propagation rate constante = initiator efficiencyk dK t = termination rate constantVV,,E

= initiator decomposition rate constant

= volume of the reaction mixture= initial volume of the reaction mixture= shrinkage proportionality constant (often con-

sidered = (p,, p M ) / p p ,where p.,, and p,. arcdensities of monomer and polymer, respec-

tively.

Whe n diffusion control begins, k,,beco mes a functionof time. Fur thermo re, t he form of the right-hand side ofE 4 1 can change if the reaction mechanism changes(e. g. , primary radical termination becomes important)or if the stationary state assumption is violated.

Therefore, assume that, at any conversion, suchnonidealiti es can be ac counted for by addition of agene ral function of time and initial initiator concentra-tion to the right-hand side of E 4 1 after all the conver-sion terms are transposed to the left-hand side.

The n, showing the integration to any conversion, wehave:

where

f = cpliLc,,)and let

Then, after integration, followed by expansion of theexponential in Taylor Series and retention of only thefirst two te rms (usually a good approximation and wellworthwhile, since it groups k d with oth er unknown pa-rame ters in a new para mete r K-thus avoiding th e nee dto independently estimate a value for kd), we obtain:

778 POLYMER ENGINEERING AND SCIENCE, AUGUST,1982, Vof. 22, No. 12


3/11

Conoersion Prediction in H i g h Conversion Free-Radical Polymerizations

A = K t + F (4)

K = k,c# ( 4 4where

and

For the special case of E = 0.

A = - ln ( l x ( 4 4

F = O ( 5 )

Low Conversion Analysis. At low conversion,

The constant K in E q 4 is deter min ed by using only thelow conversion data and a single variable search.

High Conversion Analysis. At high conversions, ki-netic similarity with a known reference conversionus. time curve is assumed. Only this one referencecurve is fit by empirical or mechanistically-based func-tions (thi rd-or der splines are used in this study) so thatthe reference conversion can be readily calculated atany time.

Therefore, using - above symbols to indicate therefercnce:

F = ; i - 6 )

Now, for any other similar polymerization, assume:

( 7 )

where the subscript L refers to the value calculatedusing limiting conversions.

The meaning of E y 7 can readily be understood byreference to Fig. 1 (which shows the reference curvedimensions), and rearrangement of this equation togive -

F - F 8

FL Fl.

It can be seen that, by kinetic similarity, here is meantthat the ratio of th e two indicated vertical distances F : F Lare equal to those of the reference curve.

In general, for E q 8 to be valid, th e time values of thereference curve must be scaled to new times, t . In theremainder octhis development, it wgl be implicitly as-sumed that F is always evaluate d at t. Then the two es-sential related problems become choice of the refer-ence curve a nd choice of a scaling function to relat e t tot . A reference curve displaying the frequentl y observedsigmoidal shape, and preferably for the polymer ofprime interest, is a reasonable first choice.

Th e functional relationship ne ede d betw een t and t ismore difficult to anticipate.

Two functional forms are con sider ed here , involvingone and two unknown parameters, respectively.

-

- -A * A L

1.80

1.60 -

-

1.40 -

1.20 --A

1.00 -

0.80 -

0.60 -

0.40 -

0.0 100 200 300 400-t

10

Fig. 1 , Churuc te r i s t i c ctirlje f o r the reference h o n i o p o l y i i i e r i -zutioii.

In each case, after the form is assum ed, th e objectiveof th e analysis is to associate the unknown parameterswith a physical meaning. The more successful this latterstep, th e more mechanistic can be the model. From apractical viewpoint, the more mechanistic the model,the easier an d more reliable become correlation of theparameters.

One Parameter Model-lncreased Kinetic Similarity.Equation 7 can be readily rearranged to provide:

-1

Kt = --= A/, &I,) aK K

where

(9)

Now if, as an additional similarity requirement withthe ref erence, we assume that a is constant, the n it be-comes the only unkonwn param eter in determining t , tosatisfy E q 7.

SincedF dF

dtx

and if we consider

where ti is the value of t a t t = 0.

and lettingApplying the L eibnitz formula to differentiate E q 12

- - -f = K& (13)

f =K,ct (14)

and

-where K Dand KDare diffusion-controlled rat e constantsand possibly n = r = 112, we obtain

POLYMER ENGINEERING AND SCIENCE, AUGUST,1982, Vof. 22, No. I2 779


4/11

S. Bulke , L. Gurciu, arid R . Pu te l

Once a is determined by a single variable search tosolve the algebraic E9s 4 , 7 , and 9 simultaneously, E q15 implies that it will obey an Arrhenius relationshipwith respect to Temperature.

In K D = -111 ac; + constant, = ~ , ~constantz(15a)

Two-Parameter Model . Assume that the relationship

This is,

RT

between t and t is linear:

t = n + b t (16)

The parameter u can be used to define the-onset ofthe ge l effect. At low conversions, for F and F to both bezero and yet have the same initial conditions:

- Kt= , t

K

When tljs equation becomes invalid, it is becauseFand/or Fa r e non-zero. From that point onward, Ey 16

relates t and t. Therefore, define the intersection ofE 9 s 16 and 17 as the onset of the gel effect:

where t,, = value of t at the onset of the gel effect.

tion during the gel effect. From E q 16.The parameter b is related to a relative rate of reac-

d t- b -

tdX d x

If we assume that this rate can he expressed as first or-der with respect to monomer, and employing a diffu-sion-controlled rate constant, K D ,and K ), in each case,then h would be expected to obey an Arrhenius rela-tionship to T:

In b = In k,+; + constant, (19a)

E d , ~ + constant,HT

Copolymerization

Formulation of the R a t e Equat ion. Assuming shrink-age during polymerization to be negligihle, the rateequation or copolymerization has been shown to be (20):

MI M 2M I , MZ, 0

In 3 In kZ1- j3kZ2) f . d t (20)where the subscripts , andmonomer andM = concentration of monomerM,, = initial monomer concentrationk = propagation rate constant

R .

refer to each respective

=total concentration of free radicals

= k, , k22ki i

Now, let R . be the sum of the low conversion valueand an unknown time function g = (p2 (t,c,,,~ 1 . 0 ) )

ekd, exp(-k,t) ] f g = [ K twhere the parameters on the right-hand side of theequation have the same meaning as in homopolym-erization.Since

and

Then, as in the analysis of homopolymerization, as-sumin g the Taylor S eries expansion to avoid literaturekd values a nd integrating, we obtain:

Let

(25)

Low Conversion Analysis. At low conversion,

G - 0 (26)

tc I tcI.0 (27)

and

Then

Low conversion data can then be fit by a single varia-ble search for the quantity 6/(1 0).

High Conversion Analysis. At high conversions, E y26 and 27 are no longer valid. Furthermore, in copo-lymerization, composition provides a degree of free-dom not present in homopolymrrization. Physically,composition affects the rate of reaction in two primaryways:

The cornposition of the polymer produced inter-acts with molec ular weight (a nd other microstruc-ture, such as sequence length) to define theviscosity, o r (more generally) the microen-vironment for reaction. Therefore. i t directlyaffects diffusion control.Since the propagation rate constants are usuallydifferent for the different monomers present, re-action rate can be affected b y the compositiondrift of the residual monointrs in the reactionmixture.

Two ways of applying the analysis to coploymers areused in this paper. They are as follows:

780 POLYMER ENGINEERING AND SCIENCE, AUGUST, 1982,Vol. 22, No. 72


5/11

Co n v e rsio ti Prediction i n High C o noersio i Free-Radical P o l y m e rizatio i.

Homopolymer A na logue .E q 24 is rearranged to pro-vide the same form as E y 4

(29)

where

and A can be given b y E q 4 b or 4 c .Now, for kinetic similarity

I = - - L

d 1,

As for homopolymerization, the subscript l , refers toth e value a t limiting conditions (of both x and w I for co-polymers) and - above the symbol refers to the refer-ence \?due. The equatcon is assumed to be valid onlywhen rl is eyaluate_d at t . If a homopolymer referenceisused, then d and G L are replaced by F and F L , respec-tively.

Again, the two outstand ing problcms ar e choice of areference_ conversio n v s . time curve and of a functionrelating t and t. In this paper, the same homopolymerreference a s previously used was used here. Also, thetwo-parameter function for t was used Ey 16).

Separa t ion of Concers ion cind Compos i t ion Con t r i -bu t ions to React ion R ate . Rearranging E q 24 to obtain:

where

and the other variables have previously been definedE y s 29 and 25.Assume

(32)G = - G1.

GL

As before, G and C are evaluated at a new time, t .Therefore, to fit E y 31 to experimental data requires athree-paramete r search for P , a , and b (using E q 1 6 ) .

EXPERIMENTA L AND COMPUTATIONALPROCEDURES

Experimental data were obt ained from the following

Methyl methacrylat e (MMA), 45C (30)Methyl methacrylate (MMA), 50, 70, 90C (4)Ethyl methacryla te (EM A), 70, 90C (31)Vinyl Chloride (VC), 30, 50, 70C 17)Acrylonitrile (AN), 40, 60, 80C (32)Styrene Methyl Methacryl ate (SMMA), 60C (22)Styrene Acrylonitrile (SAN), 60C (23)

A Data General ECLIPSE compu ter was used for thecalculations. Cubic spline fits (32) utilized the IMSLLibrary (subroutine ICSVKU). Single variable searc hesused the Fibonacci Method. Two- and three-variablesearches used the N elder Mead version of the Simplex

sources:

Technique. Transformations were used to permit un-constrained optimization search methods while actuallyconstraining values of parameters to physicallyrealizeable values (33). The objective function used forall searches utilized least squares weighted by thesquar e of the reciprocal of the experiment al value. For atwo-variable search for a an d b Ey 16) o satisfy Eys4 and 7 , for example, the objective function was:

n 1 and n2 were chosen to focus the search on the non-ideal region while avoiding undu e weighting by eithervery high or very low conversion data. This subjectivitywas necessary to avoid drastically false optima. For theone- an d two-variable searches, a wide toleranc e for n1and n2 values was experienced. Three-parametersearches were more sensitive.

Once the parameters a and b, or the single pa-rameter a were obtained, they were used togetherwith the low conversion value of the rate constant andthe spline fit to the reference in-Eq 4 to 4escribe thewhole conversion range. Since F =-0 at t = 0 whenvalues of t < 0 were encountered, F was s e t equal tozero. Also, for expediency, linear regression w a s usedto e stimate Arrhenius plots.

-

RESULTS AND DISCUSSION

Establishment of Reference Conversion Curve

Data for methyl methacrylate at 50C and 0 .3 wt.%AIBN were chosen as reference. It is representative ofthe family of sigmoidal conversion curves. Also, be-cause of wide usage of these par ticular da ta as illustra-tive of the gel effect, methyl methacrylate MMA )wasconsidered the monomer of primary interest. Thesedata were fit by cubic splines. The coefficients de-fining the fit are shown in Table I and t he curve itselfis that not ed as 50C 0. 3 wt.% AIBN in Fig . 2 . It wasexpected that th e other MMA data would be most easilyinterpreted, while polymers drastically different fromMMA (notably, acrylonitrile (AN) an d st yrene acryloni-

tril e (SAN)) would be most difficult.

Methyl Methacrylate Polymerization

Results are shown in Table 2 . The fits obtained areshown in Figs. 2 through 5 . Figure 6 shows Arrheniusplots for thc low and high conversion data. (Note thedifferent scales for the ordina tes.)

The equation for the low conversion line is:

KCd

In k,, = In = 9692 lo3 + 25.1664T (34)

Both the one-parameter and two-parameter scaling

equations E q s 9 and 16 provi ded good descriptions ofthe data. Results of the former are shown in Table 2 andFig. 6. Limiting conversions were obtained from free-volume theory. [For polymers other than MMA, limit-ing conversions were considered to be the final datapoint, or wer e esti mated from neighboring polymeriza-

POLYMER ENGINEERING AND SCIENCE, AUGUST,1982, Vo l . 2 2 , No. 12 78 1


6/11

S. Bulke, L . Gurciu, u n d R . Pate1

Table 1. Spline Fit of Reference (MMA, 5WC, 0.3% AIBN)

C, CZ c, Y f K

,1341 674000E-02 5705650000E-05 ,2122728000E-07 .5726553000E-03 ,2286902000E03,2062531 000E-02 ,885867300E-05 . l 18474000E-05 ,2617391 000E-00 ,2832871 000E03.1303177000E-01 .1920600000E-03 - 1570543000E-04 .5827780000E-00 .2960249000E03.1027992000E-01 - 4080944000E-03 .5556075000E-05 .7474772000E-00 .3205337000E03.2884020000E-03 .4243788000E-06 - 7030664000E-08 ,8360874000E-00 .4605060000E03

A = - t,-,

x = ((C:A + C2)*A + C , )* A + Y

F o r t 460.506

x = 0.8629899563 + 0.0000054289' t

tions if the run was not carri ed to "completion." Valuesare listed in t he respective tables.] The values of limit-ing conversion obtained for M M A are satisfactory ex-cept at 45"C, where the value appears high.

t

Fig . 2. Spline f i t of MMA 50 , 0.3 w t . A l B N datcr ( r a t / nrocfelappl icut ion to o t h e r hlMA 50C tlufu i i s i i i g single-pnrcriitetersculirrg equut ion, A -0.3 u;t.%, 0 -0.391 w t . , -0.5 u t .A I B N .

0.90

0.80

0.70

0.60

X0.50

0.40

0.30

0.20

0.10

0.0 15 30 45 60 75 90 105 120 135

Fig . 3. Model cipplicatioiz to M M A 70C datcc iisirig s i t i g 1 e - p ~ -ru ms ter sccdiirg eyr ic t t io i i 0 -0.3 u ; t . , -0.5 w t . 111BAi.

Table 2. Results of Model Application t o MMA

T CC K X,.( C) moles/l . lo3 YO ff ffc,.0274'45454545454550505070709090

0.006250.01 250.02500.05000.10000.2000.01 710.02230.02850.01 710.02850.01710.0285

0.37180.52940.78631.06711.68542.26061.06431.22301.36466.18017.7685

28.541035.5000

85.6385.6385.6385.6385.6385.6387.0487.0487.0491.8991.8996.0296.02

1.01281.02741.05221.05761.1 0601.10571 oooo1.00761.01170.85300.86510.70050.6991

1.16401.1 5851.16421.14811.17811.15561.11801.11831.1153,9536,9537,7831,7707

From t he 45C data, CY was found to be related to c,,through

a c ; 0 . 0 2 7 4= constant (33)

This quantity was found to be described by (ref. Table 2and Fig . 6):

The Activation E nergies for low and high conversionresults are then 19.26 and 1.99 Kcal/mole, respectivel y.This is in accord with expectations for the conventional

t

Fig . 4 . Mo d el uppliccitiort to MhfA 90C d u t a i r s i i ig s in g l c -p ( i -rcinwtcr .r.cciliiig eytcctt iori 0 -0.3 u;/. , W - 0 . 5 ict. A1R.V.

782 POLYMER ENGINEERING AND SCIENCE, AUGUST,1982, Vol . 2 2 , No. 12


7/11

Comversion Predict ion i n H i g h Conoers ion Free-Radical Polytner i , u i o n s

-

0.90 I

0.80

0.70

0.60

0.50X

0.40

0.30

0.20

0.10

0.00 0 10 20 30 4 0 5 0 60 70 80 9 0 100

t x 1 0 - 1

F i g . 5. Model upp / i c c r f i o i ito 45C clutn uuiiig siiigle-pcirui?ietcrs c d i r i g equutiori. A 0.2 nio les A I B N I I ,+ 0 . 1 nio/e.vAIBNII, 0AIBSII . 0.0062,5 i i o l e . ~ IB XII .0.05 l l lOle.5 A I R S I I , 0.025 )1101f?.S AIB SII , 0 . 012*5 IJlOle.7

K O

- 0

-0.1

I.

0.90

0.80

0.70

0.60

X0 . 5 0

1.5. O k

-

-

-

-

-

2.5

.& ko3.0 -

3.5 -

4.0 -

4.5 -

0.20

0.1

kinetics of t he low conversion region an d for th c diffu-sion control of the higher conversions.

Ethyl Methacrylate Polymerizations

Results of E M A are very similar to thos e for M M A .They are summarized in Ta b l e 3 and th e agreement ob-tained is shown in Fig . 7.

Vinyl Chlorid e Polymerizations

Low conversion analysis was, as for the other poly-mers, straightforward. However, in t he high conversionanalysis, it was found tha t only th e two-parameter "scal-ing" equations provided good descriptions of the dat a.The only abscissa available for the conversion curveswas the prod uct of time and A .his was used in the

Table 3. Results of Model Application to EMA

70 .0200 7.745 92.10 ,9323 1.03787 0 .0500 1 1.82 92.10 .9347 1.014790 ,00320 13.762 99.23 .7366 ,8622190 ,00980 23.530 99.00 ,7212 ,81868

analysis as though it were time alone (c , , = 1). Resultsobtain ed are shown in Ta b l e 4 along with the fits to thedata (Figs. 8 and 9). The resulting parameters showedArrhenius relationships (Figs. 10 and 11):

(37)8.784 . 103

Tn K = 27.14

ti633 . 103T

n h = 32.51

Int o =

-29.78 +9.219 . lo3

T

(38)

(39)

From these equations, it appears possible that K andb have the same activation energies (17.45, 17.15Kcalimole) within experi mental e rror . This may be re-

0 90( A )

0 6 0

0 0

0 4 0

0 3 0 : /',/

X -

0 2 0 /y

~ ~ ' ~ ' ' ' ~0 0 10 20 30 40 50 60 7 0 80 10 20 30 40 50 60 70 80 90

J

POLYMER ENGINEERING AND SCIENCE, AUGUS T,1982, Vol. 22 , No. 12 783


8/11

S Bulke, L. Garcia, m d R . Patel

Table 4. Results of Model Application to Vinyl Chloride

T( C) K X L a b t,,

30 0.1622 92.92 177.5 51.68 1.76250 0.9096 92.67 156.6 387.9 0.33570 4.771 97.67 154.5 1415 0.0503

lated to the dominance of transfer to monomer as thetermination step. Also, since polymer precipitation isknown to occur in this system, the values calcul ated forthe onset of the gel effect, t,,, likely reflect polymersolubility.

Acrylonitrile Polymerizations

Again, at high conversion only, the two-parametersearch results were acc eptable. Very good fits were ob-tained to a wide variety of rate curves (Figs . 12 to 1 4 ) .However, in this case, the parameters reflect manycomplexities in the polymerization, an d the polymer it-self is very remote from the MMA refer ence used . The

Fig . 9. Model cippliccitiori u i r i / / ch lo r ide , two-purciriieter scditigeyt ici t iort , 5 0 C , 0 70C.

b

0.90

0.80

0.70

0.60

X0.50

0.40

0.30

0.20

0.10

0 0I

I 1 I I

i 12 24 36 48 60 72 84 96 108

parameter s are listed in Table 5 . A reasonable approachfor this a nd similar difficult systems is to choose a refer-ence curve of the same polymer.

Copolymerization

Average composition data accompanying conversionfor copolymerization arc inherently more uncertainthan the latter because of measurement difficulties andbecause of the increased complexity of copolymeriza-

tion us . homopolymerization (21-23). Composition datahave been observed to display irregular behavior athigher conversions. Even conversion data ar e less relia-ble than for homopolymerization because of differingpolym er solubilities during precipitation methods usedin its experimental measuremelit.

784 POLYMER ENGINEERING AND SCIENCE, AUGUST, 1982 ,Vol . 22, No. 12


9/11

Con ver.yion Predictiori i r i High Coiice rsion Free-Radical Pol lnierizcitioris

X

0.0 12 24 36 4 8 60 72 84 96 108

0

0.80

0.50

0.40

0.30

:/.-.-.-.-.-.-

X

0.0 1 I I I I I I I 10.0 60 120 180 240 300 360 4 2 0 480 5 4 0 600

Copolymer data of styrene methyl methacrylate andstyrene acrylonitrile were examined using both theHomopolymer Analogue approach ( E q s 29 and 30)and the Separation of Conversion and Composition

Table 5. Results of Model Application to Acrylonitrile

T C,, K(C) . 108 103 X L a b

404040406 06 06 06 080808 0

9.82624.649.1398.26

0.4911.2282.4573.6850.2460.4911.228

0.31270.41911.1242.11130.26790.61831.17131.96100.97072.9295.450

88.00 209.088.20 200.787.50 191.1

92.00 133.593.40 131.092.80 135.421 oo 70.3990.00 118.894.50 117.7

0.09420.21 170.4340

0.1 7490.40570.60050.971 70.38182.3738

POLYM ERENGINEERING A ND SCIENCE, AUGUST,1982, Vo l . 22, No. 12

Contributions (Eys 31 and 32). The former methodreadily provided good fits to the data. In the lattermethod , however, the irregular behavior of the compo-sition values was reflected in the conversion us . timevalues calculated by the metho d. This was particularlysevere for the styrene methyl methacrylate data. Theresults for the styrene aerlonitrile data were more ac-ceptable a nd are shown in F ig . 15 for both approaches.It was found for these data that smoothing the composi-tion values by using the copolymer equation removedthe irregularities in the results but at t he expense of aless adequate fit. In general, for the case of copoly-mers, it is evident that more attention must be paid tothe problems of smoothing and parameter estimationbefore useful correlations of these values can be ob-tained.

CONCLUSIONS

A definition of kinetic similarity for diffusion-controlled polymerization reactions was intro-duced and shown to be useful for both homopoly-merizations and compolymerizations. This wasaccomplished by incorporating it into a generalapproach that permits use of the usual kineticmodelling assumptions at low conversion butavoids them in the higher conversion region,where their validity is more questionable.The approach, which we now suggest be termedthe Method of Kinetic Similarity, w a s appliedwith a single MMA, 50C kinetic curve as refer-ence throughout. This on e reference curve is the

only one that required an initial empirical (ormechanistic) fit as a prelude to the use of thismethod. A third- order spline fit was used and isprovi ded. This fit can be used for new polymeri-zation systems.For MMA and E MA, a single parameter describesthe nonide alities in the conversion 1;s. time curve

t

Fig . 1 5 . M o d e l cipplicafioii to styr-eiie cicryloiiitrile, t tco-puruni-e t e r scu l i i i g eqicutiori. L L ~ . ~ 0.5 rnolur (a), 0.7 niolcir 31, 0.9nio lur ( 0 st


10/11

S. Balke , L. Garc ia , and R. Patel

over the whole range of conversion. When thisone parameter is combined with the usual single,low-conversion-rate constant and a spline fit ofthe MMA, 50C refere nce curve, it is shown thatMMA data from 45C to 90C an d EMA data o f7 0and 90C can be well-described. Furthermore,the parameters show expected Arrhenius depen-dencies, which imply that the model is suffi-ciently mechanistic to be useful for prediction.For VC and AN, two parameters are needed todescribe the nonidealities and to provide gooddescriptions of the data. For VC, these parame-ters can be inter prete d in terms of Arrhenius de-pendencies. AN results are more complex, and aninterpretation has not yet been attempted.Copolymerizations SMMA and SAN were also ex-amined with two-parameter and three-para meternonideality descriptions. Description of thedata by the model appears to be acceptable, butirregularities in experime ntal composition valuesand the ad ded dimension of the parameter esti-mation problem require more attention.

RE CO MME N D AT I O N S

Only one reference curve used for a wide varietyof polymers and copolymers. Use of a referenceidentical to the polyme r of inter est might providesufficient kinetic similarity for th e n ee d of onlyone parameter for nonideality description regard-less of the polymer considere d.Statistical evaluation of the mode l within t he con-

text of the experimental errors associated withconversion a nd composition meas ureme nts nee dsto be explored. This is needed to provide in-creased reliability in the parameter correlations.Such an evaluation is currently in progress.

N O ME N CL AT U RE

= function of E and X (Eys 4 h and 4 c )= function of w , and P Ey 31a= model parameter ( E q 16)= model parameter Ey 16)= initial initiator concentratio n= initiator efficiency= activation energy ( E y 15a)= acceleration function ( E q 2 a )= integrated acceleration function ( E q 3 )= acceleration function ( E y 21 )= acceleration function ( E y 24b)= integrated acceleration function Ey 25)= rate parameter defined by E y 4 a )= initiator decomposition rate constant= copolymerization propagation rate constant

= group of rate constants defined by E q l a= propagation rate constant= diffusion controlled rate constant ( E y 15a)= monomer concentration of monomer i ( E q20)= initial monomer concentration Ey 20)= model parameter ( E y 14)

(Ey 20)

O(a,b) = objective function ( E q 33)R .t = timet o

T = absolute temperatureVvow 1

w,,,,

= total concentration of free radicals ( E q 20

= time at onset of gel effect ( E q 18)

= volume of the reaction mixture= initial volume of the reaction mixture= weight fraction of residual mo nomer 1 in re-

action mixture Ey 22 )= initial value of w l Ey 2 2 )

X

PE-

L

Y4J

1.

3.

4.

5 .

7-.

6.

-1 .8.

9.

10.11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.09--_23.

24

= conversion of monomer to polymer= group of rate constants ( E q 20)= shrinkage proportionality consta nt ( E q 1b )= superscript indicating value from referenc e

= subscript indicating value using limiting

= group of rate constants ( E y 2 4 a )= integrated acceleration function ( E q 29a

curve

conversion

RE FE RE N CE SW. H. Ray a i d C. E. Gall , ,Macroiiiolecule.s, 1, 425 (1969).h l . Tirrell a i i t l Ii G r o ml e y, C h e m . K q . ci. , 36, 367 (1981).L. H. Garcin-Ruhio, J . F. XlacGregor, a n d .4. E . Haniielec,.4CS Natiotial hlevting, K e w York, Aug. 1981.S. T. Balke and A . E . Hamielec, J A p p l .P o l y n i .Sci. , 17,905(1973).S. T. Balke and R. I . Patel, High Conversion Polyiieriza-tioii Kinetic hlodeli1ig Utilizing Ccl Perme:ttiolr Chroma-tography, i n .4CS S iirposiunr Series 138, T . Pi-ovtler. ed.,American Cheniicnl So ciety, Washin gton, I X 1980).K Horie, J . hlita, a i i d H . Kaml)e, J P o l y m . Sci . ,A-1 , 6,2663(1968).R. T. Ross, r . and R . L. I,aiirence,AIChE S y t t i p . Series N o .160, 72, 74.F. L. llarteir antl A. E. Hamielec, High CoiiversioiiDiffrision-Coiitrolleti Polynierizatioti, iii ACS Spposiiuii iSerirs , 104, J . N . Henderson aiid T. C. Boutoii, ed s . , Anieri-can Cheinic al Societ , Washin gtoll, DC (1979).J . h i . Dionisio a ~ i c l . . Ollriscoll, J . P o / ~ / t r i . ci. , 18, 3199(1980).E;. Arai arid S Saito, J C h c n i . Et ig . o f J upu t i , 9, :302 (1976).J. N . Cartleirus antl K. F. ODriscoll, J PO/ IPII.ci., 14, 883(1976).J . N . Cardeiias and K. . ODriscoll, J . Poly711 . ci . , 15, 1883(1977).J. 11. Dionisio and K. F. OIlriscoll, J P o / / t t i . Sci . , 18, 241

(1980).A . Ahnin and E. A. Lissi, J Alacromol . S c i . , 4 h e r i i . , A l l ,287 (1977).M. . Lachinov, R . A . Siinonian, T. G . Georgieva, V . P.Ziibov, and V . A . Kahanov, J . Polym. Sci., 17, 613 (1979).A . Hui aiid A . E. Hnririelec, J . Appl . Polyni . Sci., 16, 74911972).A. Alxle l Alim antl A. E. Haiiiielec, J . Appl. P o l y r n .Sci., 16,783 (1972)).T. Ishige a i i t l .4. . Hmiielec, J . Appl. P o l / u t .S c i . , 17, 1479(1973).L. H . Garciii-Rubio and A . E. Haiiiielec, J A p p l . Pohyrri.Sci . , 23, 1413, (1979).K . F. ODriscoll mid R. Kiiorr, ~~\ . l t rc .~-onioleculas,, 367,(1968).

hi. Johnson, T. S . K m i i o , a i i d R. R. Sniith,Eur. P o l y t t i . J . ,14,409 (1978).J. h l . Dionisio m t l K F. Ollriscoll, J Polyni . Sci., B , 17,701(1979).L. 11. Garci;i-Ruhio, Ph.D. Thesis, hlchlaster University198 1).

T. Iulig a i i t l X I . Tirrell, Xlncro?r iolec~rles,4, 1501 (1981).

POLYMER ENGINEERING AND SCIENCE, AUGUST,1982, Vol. 22, No. 12


11/11

Conoersion Prediction in H i g h Conver.sion Free-Radical Polymerizations

25. S. K. Soh and D. C . Sundberg, J . Pol /n i . S c i . , 20, 1299(1982); 1315 (1982), 1331 (198 2), 1345 (1982).

26. H . K . Mahabadi atid K. F. ODriscoll, Makromol . Chern.,179, 1921 (1958).

27. J . A . Biesenberger and R. Capinpin, J A p p l .Polym.Sci . , 16,695 (1972).

28. W . H. Lawton, E . A . Sylvestre, and hl . S. hiaggio, Te c h n o -tnetr ics , 14 513 (1972).

29. R. B. Bird, R. C . Arnistrong, a n d 0 Hassager, Dyriamics of

Polymeric Liquids, Voluriies, Fluid hlechaiiics, J o h nWiley & Sons, New Yo r k (1977).

30. K. Ito, J . Polyni. Sci., 13, 401 (1975).31. J . N. Cardenas and K. F. ODriscoll, J . Po/ yn~. ci., 15,2097

(1977).32. L. H . Garcia-Rul)io, h l . Eng. Thesis, Xlchiastcr Uiiiversity

(1975).33. J . Kowalik and M. R . Ohboriie, hlethods f o r Uiiconstriii~~ed

Optimization Probleiiis, Americaii Elsevier Pub l i sh ingCo. , New York (1968).

conversion prediction in high conversion free-radical poiymerizations

Documents