Online Polymerization Monitoring in a Continuous Reactor · Online Polymerization Monitoring in a...

586

Online Polymerization Monitoring in a ContinuousReactor

Bruno Grassl,a Wayne F. Reed*

Tulane University, Physics Dept., 2001 Percival Stern, 6400 Freret St, New Orleans, La., 70118, USAFax: +1 504-862-8702; E-mail: [email protected]

Keywords: continuous polymerization reactor; light scattering; online monitoring; polymerization; polymerization kinetics;

IntroductionContinuous reactors are commonly used for producingsynthetic polymers. In many cases they offer certainadvantages over batch reactors in terms of product qualityand ease of handling reagents and product. Because reac-tions can reach a steady state in continuous reactors, thisapproach can also be of fundamental value in studyingkinetics and mechanisms of reactions. A substantial litera-ture exists concerning the modeling of such reactors in dif-ferent contexts. A good overview is given by Dotson etal.[1] Simulations in continuous reactors have been carriedout for long chain branching,[2] copolymerization,[3, 4]

emulsion polymerization,[4] living polymerization,[5]

multi-component chain growth reactions,[6] and for variousother aspects of specific polymers such as nylon 6,[7] poly-(methyl methacrylate),[8] poly(vinyl acetate) (modelingand experiments),[9] and polystyrene.[10] Scale up modelingto full scale industrial reactors has also been made.[11]

An automatic, continuous online method for monitor-ing polymerization reactions was recently introduced.[12]

A small sample stream is continuously withdrawn fromthe reactor and diluted with solvent, using a mixingpump, in order to produce a dilute polymer solutions thatis pumped through a chain of detectors, comprisingmulti-angle time dependent static light scattering(TDSLS), ultraviolet absorption (UV), refractive index(RI), and viscosity. These detectors permit the continu-ous, online determination of weight average molecularmass M

—w, monomer conversion f, z-average mean square

radius of gyration pS 2Pz, and viscosity. It was recentlydemonstrated that these data also furnish several meansfor online estimation of the evolution of polydispersity.[13]

The technique also provides a detailed means for follow-ing polymerization kinetics,[14] and determining chaintransfer constants.[15] Reports have been published on freeradical polymerization of poly(vinyl pyrrolidone)(PVP),[12] and polyacrylamide (PAAm),[14] and for thestep growth synthesis of polyurethane.[16]

Full Paper: A recently introduced online monitoringtechnique allows monomer conversion f, weight averagepolymer mass M

—w, and reduced viscosity gr to be continu-

ously monitored without any chromatographic columnsduring polymerization reactions. This technique wasadapted to a Homogeneous Continuous Stirred TankReactor (HCSTR) to verify the quantitative predictionsconcerning f, M

—w, and gr, as a function of the flow and

kinetic parameters, to determine the kinetic parametersthemselves, to ascertain the ideality of mixing in the reac-tor, to assess the effects of feed and reactor fluctuations,and to approximate a fully continuous tube type reactor.The synthesis of polyacrylamide was chosen as a practicalsystem for the investigation. The online method should bea useful experimental technique for basic kinetic studies,development of new materials, assessment of reactor per-formance, verification of model studies, and monitoringof industrial scale reactors.

Macromol. Chem. Phys. 2002, 203, No. 3 i WILEY-VCH Verlag GmbH, 69469 Weinheim 2002 1022-1352/2002/0302–0586$17.50+.50/0

a On leave from University of Pau, France.

Schematic diagram of HCSTR and online monitoring system.

Macromol. Chem. Phys. 2002, 203, 586–597

Online Polymerization Monitoring in a Continuous Reactor 587

All these previous applications of the online methodinvolved polymerization in batch reactors. The purposeof this work is to extend the applicability to free radicalpolymerization in Homogeneous Continuous StirredTank Reactors (HCSTR). Practical issues in the operationof continuous reactors include the time for steady stateconditions to be reached, stability of the steady state, andthe conditions of polymerization occurring in the steadystate. The free radical polymerization of acrylamide(AAm) to produce polyacrylamide (PAAm) was chosento demonstrate the method.

In an HCSTR in which monomer and initiator are fedinto the reactor at the same flow rate r, at which materialis removed, a steady-state condition is reached in whichthe reactor contents will remain at constant values of M

—w,

conversion, polydispersity, etc. Here, we first demon-strate the feasibility of making continuous online mea-surements on an HCSTR, then examine the kineticapproach to the steady state, and the relation betweenpolymerization rate constants and r/V (where V is thereactor volume). The relationship of M

—w and reduced

viscosity [g] is then considered, as well as instantaneousvalues of these quantities during the transition betweensteady states. The ideal relationships among these quanti-ties require perfect stirring of the reactor. It is hencedemonstrated that deviations from perfect stirring canalso be monitored online.

It is hoped that the online monitoring technique willprove valuable both at fundamental levels of investiga-tion, where kinetic data and polymer characterization canbe concisely and economically obtained for new syn-theses and for process optimization, for verification ofmodels, and as a means of providing feedback control forindustrial scale reactors, to improve economic perfor-mance and quality control.

Polymerization and Flow ConsiderationsPrevious online monitoring of persulfate initiated PAAmpolymerization confirmed that the polymerization pro-ceeds according to ideal free radical polymerizationkinetics, at least over the majority of the monomer con-version process.[14] In this kinetic scheme the molar con-centration of monomer [m], decreases according to

d½m�dt¼ ÿ2Fkd½I2� ÿ kp½R�½m� ð1Þ

where [R] is the instantaneous molar concentration ofpropagating free radical, [I2] is the instantaneous molarconcentration of initiator, kd is the initiator decompositionrate, and kp is the polymer chain propagation rate, and Fdesignates the efficiency of initiation. The propagatingfree radical concentration obeys the rate equation

d½R�dt¼ 2Fkd½I2� ÿ kt½R�2 ð2Þ

where kt is the radical termination rate constant, themechanism for which can be via disproportionation and/or recombination. The factor of two reflects the fact thata decomposed initiator generates two free radicals I9, andR designates macro-radicals. The latter equation assumesthat the initiation step

I9 + m ggski

R (3)

is much faster than the initiator decomposition step, sothat 2Fkd [I2] is the initiation rate controlling step.

The quasi-steady state approximation (QSSA)1 assumesthat the rate of production of R is balanced by its rate ofconsumption; i.e. d[R]/dt = 0, so that

½R� X

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Fkd½I2�

kt

sð4Þ

In the long chain approximation the monomer loss inEquation (1) is due almost entirely to incorporation intothe growing polymer chain, so that the first term on theright hand side can be neglected. If the concentrationchange of initiator is negligible over the course of theentire conversion of monomer, then [I2] above can bereplaced by its initial value [I2]0, and then Equation (1)can be integrated immediately to obtain a first orderdecay of [m],

[m](t) = [m]0 exp[–kp [R] t] (5)

The monomer conversion f (t), is defined as

f ðtÞ ¼ 1ÿ ½m�ðtÞ½m�ð0Þ ð6Þ

The so-called kinetic chain length m is the ratio of theprobability of propagation to that of termination. It is theequal to the instantaneous number average degree ofpolymerization Nn, inst, when chain transfer is absent,

Nn;inst ¼kp½m�kt½R�

ð7Þ

Nn, inst is simply related to the instantaneous numberaverage polymer mass M

—n, inst by multiplication by the

monomer mass of acrylamide, MAAm.

Hence, in a batch reactor obeying the QSSA, withoutany flow of reagents, M

—n, inst will decrease as polymeriza-

tion proceeds, since [m] decreases exponentially, whereas[R] remains essentially constant. The instantaneousweight average molecular mass M

—w, inst will likewise

decrease with increasing monomer conversion, leading toan increase in polydispersity with increasing conversion.

588 B. Grassl, W. F. Reed

Similarly, reduced viscosity gr will also decrease since itis proportional to a polynomial of positive coefficients inc and intrinsic viscosity [g], the latter of which is itselfproportional to between the square root and 0.8th powerof the polymer mass.

Adapting the rate equation for [m](t) in the context ofan HCSTR is now considered. Perfect stirring is definedas follows; when an infinitesimal mass element dm fromthe monomer reservoir is introduced into the reactor overan interval dt, the probability that it is withdrawn fromthe reactor in any succeeding interval dt is identical to theprobability of any other mass element being withdrawn.Let r be the volume flow rate (cm3/s) from the reservoirto the reactor. Material is withdrawn from the reactor atthe same rate. Then, when both feed/withdrawal and thepolymerization reaction are both taking place, the instan-taneous change in the reactor in [m] is given by

d½m� ¼ rVð½m�s ÿ ½m�Þ ÿ kp½R�½m�

n odt ð8Þ

where V is the volume of the reactor, and [m]s is themolar concentration of monomer in the monomer reser-voir that feeds the reactor. Defining the mixing rate con-stant (1/s) as

p = r/V (9)

for shorthand purposes, and integrating Equation (8)yields

½m�ðtÞ ¼ ½m�r ÿp½m�s

pþ kp½R�

� �expðÿðpþ kp½R�ÞtÞ

þ ppþ kp½R�

½m�s ð10Þ

Here [m]r is the initial concentration of monomer in thereactor. It should be noted that 1/p is the average resi-dence time of a mass element in a perfectly stirred mixingchamber. Equation (10) indicates that if the initial mono-mer concentration in the reactor is the same as in thereservoir then the monomer concentration in the reactorwill exponentially decrease to a smaller value, given bythe rightmost term in Equation (10). [m] will then remainat this steady state value for as long as the reactor is fed.At this plateau value, monomer conversion f, will remainat a constant value given by

fsteady state ¼kp½R�

pþ kp½R�ð11Þ

Likewise, Nn, inst will remain constant, and can bedenoted by Nn, steady state

Nn; steady state ¼pkp½m�s

kt½R�ðpþ kp½R�Þð12Þ

The polydispersity will also remain constant, with the‘natural’ instantaneous value characteristic of the type oftermination mechanism; e.g. for termination by dispro-portionation, the instantaneous values of the n-, w- and z-averages of the polymer mass (M

—n, M

—w, and M

—z) stand in

the ratio 1:2 :3, respectively.[17]

Equation (10) also predicts that if the exponential pre-factor on the right hand side is set equal to zero, by prop-erly choosing [m]r and [m]s for given p and kp[R], thenthere will be no exponential approach to the steady stateand, in fact, the reactor will commence at and remain inthe steady state. This implies that under this condition anHCSTR can approximate the operation of a fully continu-ous tube type reactor.

In dealing with the total concentration of polymer andmonomer in the reactor it is more convenient to use massconcentration units (g/cm3). Using ct (t) to denote the timedependent total combined concentration of polymer c (t),and monomer cm(t),

ctðtÞ ¼ cmðtÞ þ cðtÞ ð13Þ

then the rate of change of ct (t), which is due entirely toflow, is given by

dct ¼ p½cm;s ÿ ctðtÞ�dt ð14Þ

which can be integrated directly, and combined withEquation (10) for [m], now in mass concentration units,to give the following expressions for ct (t), cm (t) and c (t):

ctðtÞ ¼ cm;s þ ðcm;r ÿ cm;sÞexpðÿptÞ ð15Þ

cmðtÞ ¼ cm;r ÿpcm;s

pþ a

� �exp½ÿðpþ aÞt� þ pcm;s

pþ að16Þ

cðtÞ ¼ acm;s

pþ aþ ðcm;r ÿ cm;sÞexpðÿptÞ ÿ cm;r ÿ

pcm;s

pþ a

� �exp½ÿðpþ aÞt� ð17Þ

Here cm,r is the initial concentration of monomer in thereactor, cm,s is the concentration of monomer in the sol-vent reservoir, and the polymerization rate constant hasbeen expressed as

a ¼ kp½R� ð18Þ

The following experiments both investigate and makepractical use of the above equations.

Experimental PartThe online monitoring apparatus has been previouslydescribed in detail.[12, 14] Briefly, the system works as follows:A binary mixing pump continuously withdraws a fixed per-centage (typically 3%) of reactor liquid and mixes it with the


corresponding amount of pure solvent (97%) from a separatereservoir. The diluted polymer solution (with combinedpolymer and monomer less than 0.1% by weight) is continu-ously pumped through a TDSLS detector, a UV detector, asingle capillary viscometer and an RI detector. The UVdetector (set at 225 nm) allows conversion of the acrylamide(AAm) to be monitored since, once incorporated into a pro-pagating chain the AAm double bond and its UV absorbanceis lost, so that the total polymer concentration c, at eachinstant is known. This c, together with the TDSLS signalallows the weight average polymer mass M

—w and z-average

mean square radius of gyration pS2Pz to be computed at eachinstant, according to the Zimm single contact expression inthe limit where q2pS2Pzs 1

KcIðqÞ ¼

1Mw

1þ q2pS2Pz

3

� �þ 2A2c ð19Þ

where q is the magnitude of the scattering vector

q ¼ 4pnk

� �sinðh=2Þ ð20Þ

and n is the index of refraction of the pure solvent (n = 1.333for pure water), k is the vacuum wavelength of the incidentlaser light (677 nm for the 25 mW, vertically polarized diodelaser used in the TDSLS detector), and h is the scatteringangle of detection. K is an optical constant, given for verti-cally polarized incident light by

K ¼ 4p2n2ðdn=dcÞ2

NAk4 ð21Þ

where dn/dc is the incremental index of refraction of thepolymer in the solvent, and NA is Avogadro’s number. Thevalues of dn/dc for AAm and PAAm were determined sepa-rately with a flow technique[18] and found to be 0.154 and0.193, respectively. The relationship of the PAAm value toliterature values provided by Kulicke et al.[19] The combinedRI and UV signals allow the individual values of cm(t) andc(t) to be determined at each instant.[12, 14]

A2 is the second virial coefficient of the polymer in the sol-vent, and was found to be 5.69610–4 cm3 N mol/g2, forPAAm in pure water, as determined separately.[14] This valueis used as the correction term, 2A2c in Equation (19), in thecomputation of M

—w in all the data analyses. This value was

found to be constant, within error bars, over the M—

w range ofinterest here.[14]

The home-built TDSLS detector had permanently mountedfiber optic detectors for simultaneous detection of light scat-tered at six different angles (from 35 to 1248). The devicehas been previously described in detail.[20] The UV detectorwas a Shimadzu SPD-10AV, and the RI was a Waters 410.The homebuilt viscometer was constructed from a ValidyneEngineering differential pressure transducer and a singlecapillary. It has previously been described and its perform-ance assessed.[21] The results of online monitoring have beenpreviously cross-checked successfully with gel permeationchromatography.[12, 14]

Continuous HCSTR feed and withdrawal, as well as dilu-tion were accomplished in the following manner: A multi-head peristaltic pump contained two separate plastic Tygontubes, one of which fed the reactor from a reservoir contain-ing a monomer/initiator solution, and the second of whichwithdrew solution from the reactor at the same rate r, as thefeed. The withdrawal tube contained a ‘y’-connector, onebranch of which fed the binary mixing pump, and the otherbranch of which flowed to a collection vessel. The flow rater, of the feed/withdrawal could be varied as desired, towithin 0.01 ml/min.

The flow rate of the diluted solution issuing from the mix-ing pump and feeding the detectors was set at Q = 2 ml/min.The lag time between the reactor and the detector train wasconstant throughout each experiment and was about 120 s.Figure 1 gives a schematic diagram of the apparatus.

The general scheme for the experiments was as follows:The detector train was first stabilized for several minutes bypumping pure solvent (water) from a reservoir at T = 258C.The reactor contained 40 ml of solution and normally initi-ally contained both monomer and initiator at room tempera-ture. The initial AAm concentration in both the reactor andthe reactor feed reservoir was 0.034 g/cm3, unless otherwisenoted. When the reaction was about to begin, the peristalticpump began to feed/withdraw from the reactor at a chosenrate r. The reactor was then immersed in a temperature con-trolled bath at T = 608C, and, via a thermocouple immersedin the reactor, it was determined that the reactor arrived atT = 608C in 110 s. This temperature sufficed to begin initia-tor decomposition and commence the polymerization pro-cess, which then continued throughout the rest of the experi-ment. When r was changed during an experiment, slightadjustments to the bath temperature were made to insure thereactor remained at T = 608C. The actual temperature wouldvary by as much as 18C during such adjustments.

The temperature of the detectors themselves was alwayskept at room temperature, T = 258C, except for the RI,which was set to constantly run at T = 308C via its internalheater. Data from all detectors were sampled and storedevery two seconds.

Potassium persulfate (99% minimum purity) and ultrapureelectrophoresis grade AAm were purchased from Poly-

Figure 1. Schematic diagram of HCSTR and online monitoringsystem.


sciences Inc., and used without further purification. Deion-ized water whose electrical conductivity was 18.3 lS N cm–1

was filtered through a 0.22 micron Millipore filter. A mag-netic stirring bar was used inside the reactor. The reactorwas purged with nitrogen half prior to beginning experimentsand throughout the entire polymerization reaction. Polyacry-lamide reactions have been studied in detail by other investi-gators.[22–25]

Results and DiscussionTable 1 gives a summary of the five main experimentsused to investigate the HCSTR system.

Variable Reactor Feed/Withdrawal Flow Rate

Equation (12) and (15)–(18) above indicate that conver-sion, and hence M

—w are functions of both p and a. Hence,

if the initiator concentration in both the reservoir andreactor is kept constant and p is changed (by changing thereactor feed/withdrawal rate r) intermittently, then differ-ent steady state values of f and M

—w should be obtained.

Figure 2a shows raw data for experiment #1 in whichthe flow rate was ramped from high to low values. Theinitiator concentration was 1.6 mg/ml, yielding [m]0/[I2]0

= 79.2 (M/M). Shown are the raw voltage signals for theUV, RI, TDSLS at h = 908, and viscosity, and tempera-ture (in C). Up to about 500 s pure water flowed throughthe detectors, after which 5% was pulled from the reactorand the other 95% from the pure water reservoir, for therest of the experiment. The flow rate through the detectortrain was 2 ml/min throughout the experiment. The reac-tor feed flow rate r, was 3.13 ml/min up until about t =7400 s. The RI and UV detectors reached their plateausdue to the 5% monomer flow by about t = 2400 s, atwhich point the reactor temperature was increased from T= 258C to T = 608C within 110 s. The abrupt temperaturerise in the reactor is clearly seen in Figure 2a. The hightemperature caused the persulfate initiator decompositionto begin and hence started the polymerization reaction.The exponential drop of the UV marks the approach tothe reactor steady state at r = 3.13 ml/min. The plateaureached by the UV indicates a steady concentration ofmonomer in the reactor. The increase and plateaus inTDSLS and viscosity signals show the appearance of

polymer in the reactor. T remains at 59.38C l 0.58Cthroughout the reaction.

At t = 7400 s, r was decreased to 2.66 ml/min, and,again, the exponential approach of the UV to the new

Table. 1. Conditions for HCSTR reactions. All reactions at T = 60 8C, V = 40 ml, [AAm]0 = 0.034 g/mL (except #4).

Reaction # Demonstration Figures ½I2�0=½AAm�0g=g

Monomer conversion M—

w, inst = M—

w, inst(0) + afrate kp ½R�

sÿ1

M—

w, inst(0) a

1 variable p 2a, 4–6 0.048 7.4610–4 6.86105 –6.36105

2 variable p 2b, 3, 5 0.024 5.0610–4 8.56105 –7.16105

3 variable [I2]0 7–10 0.012–0.189 variable NA NA4 approximation to tube reactor 11 0.048 NA NA NA5 parameter fluctuations 13 0.024 NA NA NA

Figure 2. (a) Raw data for experiment #1 (Table 1) in whichthe flow rate was ramped from high to low values. The numbersover the viscometer signal indicate reactor feed rate r, in ml/min. The initiator concentration was 1.6 mg/ml, yielding [m]0 /[I2]0 = 79.2 (M/M). (b) Raw data for experiment #2. The initiatorconcentration was 0.8 mg/ml, yielding [m]0 /[I2]0 = 159 (M/M).The detector voltages are in arbitrarily scaled units so that allcan be shown simultaneously. The numbers over the UV dataindicate reactor feed rate r in ml/min.


steady state is seen. Five more changes in r were thenmade; to r = 0.2.18 ml/min at t = 12000 s, to 1.71 ml/minat t = 16900 s, to 1.26 ml/min at t = 21000 s, to 0.83 ml/min at 24600 s, and to 0.22 ml/min at 28900 s. The rela-tive trends in the TDSLS and viscometer data are morecomplex than the UV data, which falls monotonicallywith decreasing r, since conversion is higher (Equation(16)). Both the TDSLS and viscometer reach maximumvalues then decrease as r decreases further and conver-sion increases further. Since M

—w from TDSLS is propor-

tional to the ratio of intensity/concentration to first order(the 2A2c correction term in Equation (19) is used in theactual M

—w computations, as mentioned), the decrease in

this ratio indicates that M—

w is decreasing with increasingconversion, a consequence of the QSSA, seen in Equation(6). The same argument holds true for the viscometer; i.e.gr, which is proportional to about M 0.75, is proportional tothe viscometer signal divided by c, again indicating, inde-pendently, that the mass is decreasing with conversion.

Figure 2b shows similar raw data for experiment #2,whose conditions are summarized in Table 1.

Figure 3 shows monomer conversion f, and M—

w, vs. tfor the data from experiment #2, where M

—w was computed

according to Equation (19).Figure 4 shows M

—w and gr vs. f for experiment #1.

Because f approaches its steady state value at a differentrate than M

—w and gr, as discussed below, there is a ‘phase

lag’ between these latter values and f, so that accuratekinetic rate data cannot be taken directly from thesecurves, although they will serve as a rough approxima-tion. Rather, it can be seen that there is a higher densityof points for both M

—w and gr when the plateau in f is

reached. The higher density of points is apparent in Fig-ure 4 because only every 20th point has been plotted forboth M

—w and gr. These values can be used for kinetic rate

determination.

Figure 3. Monomer conversion f, and M—

w, vs. t for the datafrom experiment #2 (Figure 2b), where M

—w was computed

according to Equation (19).

Figure 4. M—

w and gr vs. f for experiment #1. The points foreach steady state conversion plateau are taken from the bottomof the dense collection of points seen at intervals. Every 20th

data point was plotted in order to highlight the high point densityfrom the steady state plateau seen in Figure 2a.

(b)

Figure 5. (a) M—

w, inst taken from the plateau values of Figure 4for experiment #1 are shown vs. f. The intercept is 6.86105,and the slope is –6.306105, in good agreement with Equation(22). Similar M

—w,inst data for experiment #2 are also shown. Addi-

tionally, [g]1.33(f = 0) – [g]1.33 (f) for experiment #1 is shown,whose linear dependence on f is predicted by Equation (26). (b)log(M

—w) vs. log([g]), with the corresponding power law.


Figure 5a shows the plateau values of M—

w vs. f, forexperiments #1 and #2, along with the linear fits. In theQSSA, Equation (5) and (7) can be combined to yield

Mn, inst = Mn, 0 (1 – f) (22)

In other words, the instantaneous value of M—

n falls line-arly to zero with increasing conversion. On the steadystate plateau, when M

—w ceases to change, the detectors are

reporting the instantaneous value of M—

w (M—

w, inst) in thereactor at conversion f, which does not change as long asthe steady state is maintained. Hence, Figure 5a isequivalent to M

—w, inst (f) vs. f. Instantaneously, M

—w = 2M

—n

for termination by disproportionation, so that M—

w, inst

obeys an identical equation to Equation (22), except withM—

w,0 in place of M—

n, 0. Equation (22) predicts that the ratioof the slope in Figure 4 to the y-intercept should be –1. Infact, the ratios from Figure 4 for experiments #1 and #2are –0.93 and –0.84, which is in good agreement withEquation (22). Table 1 summarizes the fits for these data.

Figure 5a also shows [g] (f = 0)1.33–[g](f)1.33 for experi-ment #1. gr also reaches its instantaneous value at thesteady state. The relation between intrinsic viscosity andM can normally be expressed by a power law relationshipof the form (Mark-Houwink equation)

½g�w ¼ kMbw ð23Þ

where the mass average quantities have been used, sincethese are the ones measured by both the light scatteringdetector and the viscometer. The viscometer, in conjunc-tion with the RI, allows reduced viscosity to be directlymeasured, via the definition

grðf Þ ¼gðf Þ ÿ gs

cgs

¼ ð½g�ðf Þ þ j½g�ðf Þ2cðf ÞÞ ð24Þ

where [g](f) is the intrinsic viscosity of the polymer atconversion f, and j is a constant equal to about 0.35 forneutral polymers.[26] The values of [g] are on the order of500 cm3/g or less, and c (f) is less than 0.001 g/cm3 for allvalues of conversion, so that the second term on the rightof Equation (24) is small. Using j = 0.35, and solvingEquation (4) for [g], shows that the maximum correctionto gr to obtain [g], is 5%. The inset in Figure 5b shows thelog(M

—w) vs. log([g]) plot, with the parameters k = 0.015

and b = 0.75, with [g] expressed in cm3/g. While we donot expect these parameters to necessarily hold over awide range of M, since the log-log fit spans less than anorder of magnitude in [g]w and M

—w, it holds well over this

narrow range of interest.The instantaneous values of [g]w found at the conver-

sion plateaus should hence depend on M—

w,inst (f) accordingto

½g�w;instðf Þ ¼ kðMw;instðf ÞÞb ð25Þ

which, in turn, can be expressed in terms of f as

½g�1:33ðf ¼ 0Þ ÿ ½g�1:33ðf Þ ¼ ð0:015Þ1:33 2kp½m�0kt½R�

� �f ð26Þ

This latter function is also plotted in Figure 5a, wherethe predicted linear dependence is seen to hold.

Referring to Equation (16), the quantity p/(1–f) for thesteady state values of f should yield a straight line, givenby

p1ÿ f

¼ aþ p ð27Þ

Figure 6 shows this representation for experiments #1and #2. The intercept yields the first order rate constant a= kpR = 7.4610–4 s–1, and 5.0610–4 s–1, for experiments#1 and #2, respectively.

Varying Initiator Concentration at Constant ReactorFeed/Withdrawal Rate

Since the foregoing analysis has shown that it is possibleto measure both kp[R] and M

—w as reactor conditions

change, it became evident that a more complete study ofreaction kinetics, in terms of the relation between 2Fkd,M—

w, kp2/kt and [I2]0, could be made in a single experiment

by changing initiator concentration at intervals. Thisrepresents a concise, economical means of determiningreaction kinetics without the need to run several indepen-dent experiments.

Figure 7 shows M—

w and f vs. t for experiment #3, inwhich p was kept constant (7.13610–4 s–1), and theamount of initiator was increased at intervals. This wasaccomplished by switching the peristaltic feed tube into afresh reservoir containing the same concentration ofAAm (0.034 g/cm3) and the desired new concentration ofinitiator. Simultaneously, initiator was also injected into

Figure 6. p/(1–f) vs. p, according to Equation (27) for experi-ments #1 and #2. The incept yields the polymerization rate con-stant a = kp[R] for each experiment. The intercept for experiment#1 is 7.4610–4 s.1, and for experiment #2 it is 5.0610–4 s–1.


the reactor to bring it up to the level of the new reservoir,in an effort to minimize the time of approach to the newsteady state. Actually, this had the effect of transientlydisturbing the plateau. The initiator increase at t = 26000s was made only by switching to a new AAm/initiatorreservoir without changing the reactor’s initiator concen-tration. This eliminated the transient disturbance in theconversion curve.

From Equation (4) and (27) it is easily shown that thecluster of kinetic constants can be determined by plotting

½I2;0� ¼kt

2FkDk2p

!p

f1ÿ f

� �� 2

ð28aÞ

where

a2 ¼ pf

1ÿ f

� �� 2

ð28bÞ

Figure 8 shows this representation, from which theslope yields

kt

2Fkdk2p

!ðM N s2=LÞ ¼ 9075 ð29Þ

Since the behavior of both the approach to equilibriumand the equilibrium plateau itself confirm that the reactoris perfectly mixed, for all practical purposes, and that thepolymerization proceeds according to the QSSA, it ispossible to exploit the M

—w data in Figure 7 to complete

the determination of the kinetic parameters. The kineticparameter kp

2/kt can be computed from the above slope,together with the relation

dMwðf ¼ 0Þ

d

ffiffiffiffiffiffiffiffiffiffiffi1½I2�0

s ¼ MAAmkp½m�0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFkdkt=2

p ð30Þ

with MAAm = 71 g/mole, and [m]0 = 0.479 M/L, via

k2p

kt¼

12½m�0MAAm

dMwðf ¼ 0Þ

d

ffiffiffiffiffiffiffiffiffiffiffi1½I2�0

sd½I2�0da2

� �1=2ð31Þ

Figure 9 shows M—

w (f = 0), from extrapolation accord-ing to M

—w(f)/(1–f), vs. 1/sqrt ([I2]0). The value of kp

2/kt is11.7 L/M N s.

Also shown in Figure 9 are the M—

w (f = 0) points forexperiments #1 and #2, as well as the data for M

—w (f = 0)

taken from Table 1 in reference 14. All the experiments

Figure 7. M—

w and f vs. t for experiment #3, in which p was keptconstant and the amount of initiator was increased at intervals.The numbers over the M

—w data are the initial initiator concentra-

tion in 10–3m/L.

Figure 8. [I2]0 vs. [pf/(1–f)]2 according to Equation (28) forexperiment #3, including the corresponding linear fit and param-eters.

Figure 9. M—

w(f = 0), from extrapolation according to M—

w (f)/(1–f), vs. 1/sqrt([I2]0) for experiment #3. Also shown are the M

—w (f =

0) values for experiments #1 and #2, as well as M—

w (f = 0) datataken from Table 1 of ref.[14] Slope to experiment #3 is 74800.Circles represent experiment #3, squares are from Table 1 ofref.,[14] and diamonds are experiments #1 and #2.


are in good agreement, except for one point (M—

w (f = 0) L26106) that deviates significantly from the other data.

Kinetics during the Approach to Steady State

Equation (15)–(17) above indicate how ct (t), cm (t) andc (t) vary as a function of time for different combinationsof flow, polymerization and concentration conditions.

Figure 10 shows an excerpt from Figure 7 (experiment#3) of one of the exponential approaches to the steadystate. The exponential fit is quite good, in agreement withEquation (9), which confirms two important assumptionsmade: First, the reaction indeed proceeds via the QSSA,and, second, the mixing of the reactor is perfect, for allpractical purposes. This allows the rate constant to bedetermined alternatively via the rate constant of the expo-nential approach to equilibrium.

An analysis of the approach of M—

w (t) to the steady staterequires a theoretical expression. At any time t, the con-centration of ‘dead’ polymer chains in the reactor will bethe sum of the remaining concentration increments ofpolymer that were produced ever since t = 0. As notedabove, the time it takes to produce a polymer chain is farshorter than 1/p, the average residence time in the reactor.M—

w(t) is, by definition

MwðtÞ ¼

Z t

0

Mw;instðt9Þdcðt; t9ÞZ t

0

dcðt; t9Þð32Þ

M—

w, inst (t 9) is the instantaneous value of M—

w (t), and can befound by Equation (7) above as

Mw;instðtÞ ¼ 2kpcmðtÞ

kt½R�MAAm

ð33Þ

where the factor of 2 is the ratio of the instantaneousmass and number averages for free radical reactions ter-minating by disproportionation (if recombination reac-tions are also involved the factor would be less than 2),and mass concentration units are used for monomer.

Now dc (t,t 9) is the increment of concentration of poly-mer of mass M

—w, inst (t 9), produced at time t 9, and still left

in the reactor at time t. In other words, it is the product ofthe amount of polymer produced at t 9, times the fractionof polymer remaining at t,

dcðt; t9Þ ¼ qcqt

��t9;due to polymerization

24 35x½A exp½ÿpðt ÿ t9Þ��dt9 ð34Þ

Here, Aexp[–p(t–t 9)] is the probability that a mass ele-ment that entered the reactor at time t 9 is still left in thereactor at time t, assuming the reactor is perfectly mixed,as defined above. A is a normalization constant thatensures

cðtÞ ¼Z t

0

dcðt; t9Þ ¼ AaMAAm

Z t

0

cmðt9Þexp½ÿpðt ÿ t9Þ�dt9 ð35Þ

and the fact has been used that the amount of polymerproduced between the interval t 9 and t 9+dt 9 is

qcqt

��t9;due to polymerization

24 35 ¼ ÿMAAm

q½m�qt

��t9; due to polymerization

¼ MAAmacmðt9Þ ð36Þ

Integrating Equation (35), and using Equation (16) forc (t) allows A (t) to be determined,

AðtÞ ¼

acm;s

pþ aþ ðcm;r ÿ cm;sÞexpðÿptÞ ÿ cm;r ÿ

pc;m;spþ a

� �exp½ÿðpþ aÞt�

� �a expðÿptÞ cm;s

pþ a½expðptÞ ÿ 1� þ 1

acm;r ÿ

pcm;s

pþ a

� �½1ÿ expðÿatÞ�

� � ð37Þ

Using this in conjunction with Equation (35) for Mw, inst (t 9) yields

MwðtÞ ¼

pcm;s

pþ a

� �2

½expðptÞ ÿ 1� þ 2pcm;s

aðpþ aÞ cm;r ÿpcm;s

pþ a

� �½1ÿ expðÿatÞ� þ

cm;r ÿpcm;spþ a

� �2

pþ a½1ÿ exp½ÿðpþ 2aÞt��

8><>:9>=>;

kt½R�MAAm

2kp

� �cm;s

pþ a½expðptÞ ÿ 1� þ 1

acm;r ÿ

pcm;s

pþ a

� �½1ÿ expðÿatÞ�

� � ð38Þ

The case most used in this work is that cm, r = cm, s. Then Mw (t) simplifies to

MwðtÞ ¼2kpcm;s

kt½R�ðpþ aÞ

p½expðptÞ ÿ 1� þ 2p½1ÿ expðÿatÞ� þ a2

pþ 2a½1ÿ exp½ÿðpþ 2aÞt��

� �expðptÞ ÿ expðÿatÞ ð39Þ


This latter expression implies that the approach to equi-librium of M

—w(t) will always be slower than for f (t), as is

now demonstrated. Although M—

w (t) is not a pure expo-nential decay function, the time dependent portion resem-bles one, in which the initial value is 1 and the final valueis p/(p + a), as can be readily verified by taking the limitsof t = 0 and large t in Equation (39). Hence the amplitudeof the time dependent portion is a/(a + p). Now, if thelimit of small t is taken of Equation (39), by expandingthe exponentials in the numerator and denominator tosecond order in t, using the binomial expansion on thedenominator, and retaining terms to order t, one obtains,after considerable algebra

MwðtÞ L2kpcm;s

kt½R�1ÿ at

2

� �ð40Þ

Equating the time dependent part of this to the firstorder expansion of an effective exponential that has theprefactor a/(a + p) and a decay rate b,

MwðtÞ ÿ limtev

MwðtÞ ¼2kpcm;s

kt½R�1ÿ at

2

� �ÿ p

pþ a

� �

L2kpcm;s

kt½R�a

pþ að1ÿ btÞ

� �ð41Þ

yields an important approximate relationship between thedecay rate of the effective decay of the time dependentpart of M

—w (t) and the constants a and p, namely

b Xaþ p

2ð42Þ

i. e. the effective rate constant of the M—

w (t) approach toequilibrium is approximately one half the rate constant off (t) approach to equilibrium.

Figure 10 also shows M—

w (t), together with f (t), and it isseen that this relationship is closely born out. It is alsogenerally found in the other experiments for each indivi-dual parameter change leading to a new steady state.

The approach of gr to the steady state can be found byinvoking the scaling law in Equation (23). Then

½g�wðtÞ ¼kZ t

0

Mw;instðt9Þbdcðt; t9Þ

cðtÞ ð43Þ

Since b in general can range from 0.5 to 0.8 for coilpolymers, no single expression can be given for the aboveintegral. However, since it represents an average propor-tional to pMbP, we can surmise that the decay rate will belower than for M

—w(t). This qualitative observation is born

out in the inset of Figure 10, where gr (t) is plotted f, andthe pseudo-exponential decay rate is found to be 73% ofthe M

—w (t) pseudo-exponential decay rate.

Although we do not wish to pursue the details here, forcompleteness it is mentioned that the polydispersity indexM—

w(t)/M—

n(t) during the approach to the steady state can becomputed by finding M

—n(t 9) using the above method. The

number average polymer mass M—

n(t) is

MnðtÞ ¼cðtÞZ t

0

dcðt; t9ÞMn;instðt9Þ

¼ pkpcðtÞaAðtÞexpðÿptÞkt½R�MAAmðexpðptÞ ÿ 1Þ ð44Þ

where the QSSA form for M—

n, inst (t 9) = MAAmNn, inst fromEquation (7) has been used.

Approximation to a Fully Continuous Reactor

Equations (16) and (38) imply that if the initial monomerconcentration in the HCSTR is less than that in the reser-voir, and obeys the relationship

cm;r ¼pcm;s

pþ að45Þ

then cm(t) and M—

w (t) will be immediately in the steadystate as soon as the reaction begins, and the polymer con-centration will exponentially approach its full steady statevalue with the rate constant p. This would then allow theHCSTR to approximate a fully continuous tube reactor interms of cm(t) and M

—w (t).

Figure 11 shows cm (t), M—

w (t) and c (t) for experiment#4, which attempts this condition. For this experiment,p = 4.17610–4 (1/s), a = 6.69610–4 (1/s). Although a

Figure 10. f and M—

w vs. t for one of the exponential approachesto the steady state for experiment #3. M

—w approaches the steady

state in a pseudo-exponential fashion, in which the effective rateconstant is about half that of the rate constant governing theapproach of f to the steady state, according to Equation (42).The inset shows the pseudo-exponential approach of gr to thesteady state, at a rate less than M

—w (t), as predicted by Equation

(43).


perfect approximation was not achieved, due to error barsin both the experimental value of a, and in practicallyestablishing the reaction concentration and other condi-tions, it is clearly seen how greatly diminished the ampli-tudes of cm (t) and M

—w (t) are in their approach to the

steady state. The large amplitude change in cm (t) thatwould occur is seen from the data, where the system wasfirst pumped at the full reservoir monomer concentrationof cs = 0.034 g/cm3, before switching to, and stabilizingat, the initial reactor concentration of cr = 0.013 g/cm3.Although the amplitudes are greatly diminished, the rateconstants at which these residual deviations from thesteady state value relax involve the same rate constants aswhen no attempt at such a balance is made.

As expected the initial values of M—

w (t) are smaller thanat the steady state, because cm,r a cm,s and the amount ofinitiator is constant in both the reactor and reservoir at alltime. This is predicted by Equation (34), due to negativeprefactors in two of the terms.

Comparison with Batch Reactor Results

To contrast the HCSTR results, Figure 12 shows theresults for M

—w (f) for a batch reactor. The AAm initial

concentration was, 0.034 g/cm3 as in most of the otherexperiments, and [I2]0/[AAm]0 = 0.094 g/g. The inset toFigure 12 shows f vs. t. A first order fit is also shown, butis so close to the data that it is difficult to discern. Thefirst order nature of f (t) is evidence that the QSSA isapproximated by this batch reaction. The linear decreasein M

—w over most of the conversion is also characteristic of

the QSSA. The batch reactor measures the cumulativeweight average mass of all dead polymer chainsM—

w, cumulative, which, in the QSSA is given by

Mw; cumulativeðf Þ ¼ Mw;0 1ÿ f2

� �ð46Þ

This approximation is closely obeyed in the figure forconversion values above 0.2. Deviations of the PAAmreaction from ideal polymerization were discussed inreference 14.

Monitoring Fluctuations in Conversion Due to Drift inReactor Conditions

Whereas the previous experiments have shown how steadystate HCSTR operation is approached and can be main-tained, there is considerable practical interest in being ableto monitor the effects of drift in reactor conditions.

Figure 13 shows the results of experiment #5, in whicha steady state was first achieved, then the followingmanipulations made: 1) deliberate temperature fluctua-tions, 2) the effect of a change in initiator concentration,and 3) non-perfect mixing.

Figure 11. M—

w (t), cm (t) and cp (t) for experiment #4, whichattempts to approximate a fully continuous reactor.

Figure 12. M—

w (f) for a batch reactor. The inset shows f (t),along with a first order fit.

Figure 13. An HCSTR reaction in which temperature, initiatorconcentration and mixing were allowed to fluctuate. This isexperiment #5 in Table 1.


The most dramatic effects on the reaction are seenwhen the stirring rate is changed. During the last portionof the curve, stirring speed was reduced to nearly zero,then to zero altogether. The low conversion in this por-tion, and corresponding high M

—w (which follows from a

high monomer to initiator ratio at low conversion), maybe the result of monomer passing quickly from the nearlystatic reactor solution directly to the withdrawal tube,leaving little time for conversion to occur. This process ischaotic, leading to random fluctuations in the conversion.

ConclusionsThe online polymerization monitoring method has beensuccessfully adapted to a HCSTR. It has been demon-strated that conversion, and hence M

—w, can be tightly con-

trolled by reactor feed/withdrawal rate. The exponentialapproach in time of f to steady state conditions immedi-ately indicates that 1) the polymerization reaction is pro-ceeding according to QSSA (first order) kinetics and 2)that the reactor is well mixed. It has been further shownthat the method provides a concise, economical methodfor determining detailed reaction kinetics, since a widerange of initiator/monomer ratios can be quantitativelymonitored during a single experiment. It has also beendemonstrated that, by proper choice of initial monomerconcentration in the reactor and feed reservoir, the ampli-tude of the exponential approach to the steady state can besignificantly reduced; i.e. the HCSTR can be made toapproximate a continuous tube type reactor. Adaptation tocontinuous tube type reactors should be straightforward.Finally, the long term stability of the steady state wasinvestigated, as well as the effects of deliberately causingconditions to drift. In this case, deliberate fluctuations intemperature, initiator concentration and stirring rate weremade to show the deviations from steady state operation,those due to stirring being the most dramatic.

Acknowledgement: The authors gratefully acknowledge sup-port from the U.S. National Science Foundation CTS 9877206.

Received: August 29, 2001Revised: October 10, 2001

Accepted: October 22, 2001

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