Online Polymerization Monitoring in a Continuous Reactor

586 Macromol. Chem. Phys. 00, 03, 586 597 Full Paper: A recently introduced online monitoring technique allows monomer conversion f, weight average polymer mass M w, and reduced viscosity g r to be continuously monitored without any chromatographic columns during polymerization reactions. This technique was adapted to a Homogeneous Continuous Stirred Tank Reactor (HCSTR) to verify the quantitative predictions concerning f, M w, and g r, as a function of the flow and kinetic parameters, to determine the kinetic parameters themselves, to ascertain the ideality of mixing in the reactor, 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 practical system for the investigation. The online method should be a useful experimental technique for basic kinetic studies, development of new materials, assessment of reactor performance, verification of model studies, and monitoring of industrial scale reactors. Schematic diagram of HCSTR and online monitoring system. Online Polymerization Monitoring in a Continuous Reactor Bruno Grassl, a Wayne F. Reed* Tulane University, Physics Dept., 001 Percival Stern, 6400 Freret St, New Orleans, La., 70118, USA Fax: +1 504-86-870; E-mail: wreed@tulane.edu Keywords: continuous polymerization reactor; light scattering; online monitoring; polymerization; polymerization kinetics; Introduction Continuous reactors are commonly used for producing synthetic polymers. In many cases they offer certain advantages over batch reactors in terms of product quality and ease of handling reagents and product. Because reactions can reach a steady state in continuous reactors, this approach can also be of fundamental value in studying kinetics and mechanisms of reactions. A substantial literature exists concerning the modeling of such reactors in different contexts. A good overview is given by Dotson et al. [1] Simulations in continuous reactors have been carried out for long chain branching, [] [3, 4] copolymerization, emulsion polymerization, [4] living polymerization, [5] multi-component chain growth reactions, [6] and for various other aspects of specific polymers such as nylon 6, [7] poly- (methyl methacrylate), [8] poly(vinyl acetate) (modeling and experiments), [9] and polystyrene. [10] Scale up modeling to full scale industrial reactors has also been made. [11] a On leave from University of Pau, France. An automatic, continuous online method for monitoring polymerization reactions was recently introduced. [1] A small sample stream is continuously withdrawn from the reactor and diluted with solvent, using a mixing pump, in order to produce a dilute polymer solutions that is pumped through a chain of detectors, comprising multi-angle time dependent static light scattering (TDSLS), ultraviolet absorption (UV), refractive index (RI), and viscosity. These detectors permit the continuous, online determination of weight average molecular mass M w, monomer conversion f, z-average mean square radius of gyration ps P z, and viscosity. It was recently demonstrated that these data also furnish several means for online estimation of the evolution of polydispersity. [13] The technique also provides a detailed means for following polymerization kinetics, [14] and determining chain transfer constants. [15] Reports have been published on free radical polymerization of poly(vinyl pyrrolidone) (PVP), [1] and polyacrylamide (PAAm), [14] and for the step growth synthesis of polyurethane. [16] Macromol. Chem. Phys. 00, 03, No. 3 i WILEY-VCH Verlag GmbH, 69469 Weinheim 00 10-135/00/030 0586$17.50+.50/0

Online Polymerization Monitoring in a Continuous Reactor 587 All these previous applications of the online method involved polymerization in batch reactors. The purpose of this work is to extend the applicability to free radical polymerization in Homogeneous Continuous Stirred Tank Reactors (HCSTR). Practical issues in the operation of continuous reactors include the time for steady state conditions to be reached, stability of the steady state, and the conditions of polymerization occurring in the steady state. The free radical polymerization of acrylamide (AAm) to produce polyacrylamide (PAAm) was chosen to demonstrate the method. In an HCSTR in which monomer and initiator are fed into the reactor at the same flow rate r, at which material is removed, a steady-state condition is reached in which the reactor contents will remain at constant values of M w, conversion, polydispersity, etc. Here, we first demonstrate the feasibility of making continuous online measurements on an HCSTR, then examine the kinetic approach to the steady state, and the relation between polymerization rate constants and r/v (where V is the reactor volume). The relationship of M w and reduced viscosity [g] is then considered, as well as instantaneous values of these quantities during the transition between steady states. The ideal relationships among these quantities require perfect stirring of the reactor. It is hence demonstrated that deviations from perfect stirring can also be monitored online. It is hoped that the online monitoring technique will prove valuable both at fundamental levels of investigation, where kinetic data and polymer characterization can be concisely and economically obtained for new syntheses and for process optimization, for verification of models, and as a means of providing feedback control for industrial scale reactors, to improve economic performance and quality control. Polymerization and Flow Considerations Previous online monitoring of persulfate initiated PAAm polymerization confirmed that the polymerization proceeds according to ideal free radical polymerization kinetics, at least over the majority of the monomer conversion process. [14] In this kinetic scheme the molar concentration of monomer [m], decreases according to d½mš dt ¼ Fk d ½I Š k p ½RŠ½mŠ ð1þ where [R] is the instantaneous molar concentration of propagating free radical, [I ] is the instantaneous molar concentration of initiator, k d is the initiator decomposition rate, and k p is the polymer chain propagation rate, and F designates the efficiency of initiation. The propagating free radical concentration obeys the rate equation d½rš dt ¼ Fk d ½I Š k t ½RŠ where k t is the radical termination rate constant, the mechanism for which can be via disproportionation and/ or recombination. The factor of two reflects the fact that a decomposed initiator generates two free radicals I9, and R designates macro-radicals. The latter equation assumes that the initiation step I9 + m ki ggs ðþ R (3) is much faster than the initiator decomposition step, so that Fk d [I ] is the initiation rate controlling step. The quasi-steady state approximation (QSSA) 1 assumes that the rate of production of R is balanced by its rate of consumption; i.e. d[r]/dt = 0, so that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fk d ½I Š ½RŠ X k t ð4þ In the long chain approximation the monomer loss in Equation (1) is due almost entirely to incorporation into the growing polymer chain, so that the first term on the right hand side can be neglected. If the concentration change of initiator is negligible over the course of the entire conversion of monomer, then [I ] above can be replaced by its initial value [I ] 0, and then Equation (1) can be integrated immediately to obtain a first order decay of [m], [m](t) = [m] 0 exp[ k p [R]t] (5) The monomer conversion f(t), is defined as f ðtþ ¼ 1 ½mŠðtÞ ½mŠð0Þ ð6þ The so-called kinetic chain length m is the ratio of the probability of propagation to that of termination. It is the equal to the instantaneous number average degree of polymerization N n, inst, when chain transfer is absent, N n;inst ¼ k p½mš k t ½RŠ N n, inst is simply related to the instantaneous number average polymer mass M n, inst by multiplication by the monomer mass of acrylamide, M AAm. Hence, in a batch reactor obeying the QSSA, without any flow of reagents, M n, inst will decrease as polymerization proceeds, since [m] decreases exponentially, whereas [R] remains essentially constant. The instantaneous weight average molecular mass M w, inst will likewise decrease with increasing monomer conversion, leading to an increase in polydispersity with increasing conversion. ð7þ

588 B. Grassl, W. F. Reed Similarly, reduced viscosity g r will also decrease since it is proportional to a polynomial of positive coefficients in c and intrinsic viscosity [g], the latter of which is itself proportional to between the square root and 0.8 th power of the polymer mass. Adapting the rate equation for [m](t) in the context of an HCSTR is now considered. Perfect stirring is defined as follows; when an infinitesimal mass element dm from the monomer reservoir is introduced into the reactor over an interval dt, the probability that it is withdrawn from the reactor in any succeeding interval dt is identical to the probability of any other mass element being withdrawn. Let r be the volume flow rate (cm 3 /s) from the reservoir to the reactor. Material is withdrawn from the reactor at the same rate. Then, when both feed/withdrawal and the polymerization reaction are both taking place, the instantaneous change in the reactor in [m] is given by n d½mš ¼ r o V ð½mš ½mŠÞ k s p½rš½mš dt where V is the volume of the reactor, and [m] s is the molar concentration of monomer in the monomer reservoir that feeds the reactor. Defining the mixing rate constant (1/s) as ð8þ p = r/v (9) for shorthand purposes, and integrating Equation (8) yields ½mŠðtÞ ¼ ½mŠ r p½mš s expð ðp þ k p ½RŠÞtÞ p þ k p ½RŠ þ p p þ k p ½RŠ ½mŠ s ð10þ Here [m] r is the initial concentration of monomer in the reactor. It should be noted that 1/p is the average residence time of a mass element in a perfectly stirred mixing chamber. Equation (10) indicates that if the initial monomer concentration in the reactor is the same as in the reservoir then the monomer concentration in the reactor will exponentially decrease to a smaller value, given by the rightmost term in Equation (10). [m] will then remain at this steady state value for as long as the reactor is fed. At this plateau value, monomer conversion f, will remain at a constant value given by f steady state ¼ k p½rš p þ k p ½RŠ ð11þ Likewise, N n, inst will remain constant, and can be denoted by N n, steady state N n; steady state ¼ pk p ½mŠ s k t ½RŠðp þ k p ½RŠÞ ð1þ The polydispersity will also remain constant, with the natural instantaneous value characteristic of the type of termination mechanism; e. g. for termination by disproportionation, 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::3, respectively. [17] Equation (10) also predicts that if the exponential prefactor on the right hand side is set equal to zero, by properly choosing [m] r and [m] s for given p and k p [R], then there will be no exponential approach to the steady state and, in fact, the reactor will commence at and remain in the steady state. This implies that under this condition an HCSTR can approximate the operation of a fully continuous tube type reactor. In dealing with the total concentration of polymer and monomer in the reactor it is more convenient to use mass concentration units (g/cm 3 ). Using c t (t) to denote the time dependent total combined concentration of polymer c(t), and monomer c m (t), c t ðtþ ¼ c m ðtþ þ cðtþ ð13þ then the rate of change of c t (t), which is due entirely to flow, is given by dc t ¼ p½c m;s c t ðtþšdt ð14þ which can be integrated directly, and combined with Equation (10) for [m], now in mass concentration units, to give the following expressions for c t (t), c m (t) and c(t): c m ðtþ ¼ cðtþ ¼ c t ðtþ ¼ c m;s þ ðc m;r c m;s Þexpð ptþ c m;r pc m;s exp½ ðþtš þ pc m;s ac m;s þ ðc m;r c m;s Þexpð ptþ c m;r pc m;s exp½ ðþtš ð15þ ð16þ ð17þ Here c m,r is the initial concentration of monomer in the reactor, c m,s is the concentration of monomer in the solvent reservoir, and the polymerization rate constant has been expressed as a ¼ k p ½RŠ ð18þ The following experiments both investigate and make practical use of the above equations. Experimental Part The online monitoring apparatus has been previously described in detail. [1, 14] Briefly, the system works as follows: A binary mixing pump continuously withdraws a fixed percentage (typically 3%) of reactor liquid and mixes it with the

Online Polymerization Monitoring in a Continuous Reactor 589 corresponding amount of pure solvent (97%) from a separate reservoir. The diluted polymer solution (with combined polymer and monomer less than 0.1% by weight) is continuously pumped through a TDSLS detector, a UV detector, a single capillary viscometer and an RI detector. The UV detector (set at 5 nm) allows conversion of the acrylamide (AAm) to be monitored since, once incorporated into a propagating chain the AAm double bond and its UV absorbance is lost, so that the total polymer concentration c, at each instant is known. This c, together with the TDSLS signal allows the weight average polymer mass M w and z-average mean square radius of gyration ps P z to be computed at each instant, according to the Zimm single contact expression in the limit where q ps P z s 1 Kc IðqÞ ¼ 1 1 þ q ps P z þ A c M w 3 where q is the magnitude of the scattering vector q ¼ 4pn k sinðh=þ ð19þ ð0þ and n is the index of refraction of the pure solvent (n = 1.333 for pure water), k is the vacuum wavelength of the incident laser light (677 nm for the 5 mw, vertically polarized diode laser used in the TDSLS detector), and h is the scattering angle of detection. K is an optical constant, given for vertically polarized incident light by K ¼ 4p n ðdn=dcþ N A k 4 ð1þ where dn/dc is the incremental index of refraction of the polymer in the solvent, and N A is Avogadro s number. The values of dn/dc for AAm and PAAm were determined separately with a flow technique [18] and found to be 0.154 and 0.193, respectively. The relationship of the PAAm value to literature values provided by Kulicke et al. [19] The combined RI and UV signals allow the individual values of c m (t) and [1, 14] c(t) to be determined at each instant. A is the second virial coefficient of the polymer in the solvent, and was found to be 5.69610 4 cm 3 N mol/g, for PAAm in pure water, as determined separately. [14] This value is used as the correction term, A c in Equation (19), in the computation of M w in all the data analyses. This value was found to be constant, within error bars, over the M w range of interest here. [14] The home-built TDSLS detector had permanently mounted fiber optic detectors for simultaneous detection of light scattered at six different angles (from 35 to 148). The device has been previously described in detail. [0] The UV detector was a Shimadzu SPD-10AV, and the RI was a Waters 410. The homebuilt viscometer was constructed from a Validyne Engineering differential pressure transducer and a single capillary. It has previously been described and its performance assessed. [1] The results of online monitoring have been previously cross-checked successfully with gel permeation [1, 14] chromatography. Figure 1. system. Schematic diagram of HCSTR and online monitoring Continuous HCSTR feed and withdrawal, as well as dilution were accomplished in the following manner: A multihead peristaltic pump contained two separate plastic Tygon tubes, one of which fed the reactor from a reservoir containing a monomer/initiator solution, and the second of which withdrew solution from the reactor at the same rate r, as the feed. The withdrawal tube contained a y -connector, one branch of which fed the binary mixing pump, and the other branch of which flowed to a collection vessel. The flow rate r, of the feed/withdrawal could be varied as desired, to within 0.01 ml/min. The flow rate of the diluted solution issuing from the mixing pump and feeding the detectors was set at Q = ml/min. The lag time between the reactor and the detector train was constant throughout each experiment and was about 10 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 by pumping pure solvent (water) from a reservoir at T = 58C. The reactor contained 40 ml of solution and normally initially contained both monomer and initiator at room temperature. The initial AAm concentration in both the reactor and the reactor feed reservoir was 0.034 g/cm 3, unless otherwise noted. When the reaction was about to begin, the peristaltic pump began to feed/withdraw from the reactor at a chosen rate r. The reactor was then immersed in a temperature controlled bath at T = 608C, and, via a thermocouple immersed in the reactor, it was determined that the reactor arrived at T = 608C in 110 s. This temperature sufficed to begin initiator decomposition and commence the polymerization process, which then continued throughout the rest of the experiment. When r was changed during an experiment, slight adjustments to the bath temperature were made to insure the reactor remained at T = 608C. The actual temperature would vary by as much as 18C during such adjustments. The temperature of the detectors themselves was always kept at room temperature, T = 5 8C, except for the RI, which was set to constantly run at T = 308C via its internal heater. Data from all detectors were sampled and stored every two seconds. Potassium persulfate (99% minimum purity) and ultrapure electrophoresis grade AAm were purchased from Poly-

590 B. Grassl, W. F. Reed Table. 1. Conditions for HCSTR reactions. All reactions at T = 608C, V = 40 ml, [AAm] 0 = 0.034 g/ml (except #4). Reaction # Demonstration Figures ½I Š 0 =½AAmŠ 0 g=g Monomer conversion M w, inst = M w, inst(0) + af rate k p ½RŠ M w, inst(0) a s 1 1 variable p a, 4 6 0.048 7.4610 4 6.8610 5 6.3610 5 variable p b, 3, 5 0.04 5.0610 4 8.5610 5 7.1610 5 3 variable [I ] 0 7 10 0.01 0.189 variable NA NA 4 approximation to tube reactor 11 0.048 NA NA NA 5 parameter fluctuations 13 0.04 NA NA NA sciences Inc., and used without further purification. Deionized water whose electrical conductivity was 18.3 ls N cm 1 was filtered through a 0. micron Millipore filter. A magnetic stirring bar was used inside the reactor. The reactor was purged with nitrogen half prior to beginning experiments and throughout the entire polymerization reaction. Polyacrylamide reactions have been studied in detail by other investigators. [ 5] Results and Discussion Table 1 gives a summary of the five main experiments used to investigate the HCSTR system. Variable Reactor Feed/Withdrawal Flow Rate Equation (1) and (15) (18) above indicate that conversion, and hence M w are functions of both p and a. Hence, if the initiator concentration in both the reservoir and reactor is kept constant and p is changed (by changing the reactor feed/withdrawal rate r) intermittently, then different steady state values of f and M w should be obtained. Figure a shows raw data for experiment #1 in which the flow rate was ramped from high to low values. The initiator concentration was 1.6 mg/ml, yielding [m] 0 /[I ] 0 = 79. (M/M). Shown are the raw voltage signals for the UV, RI, TDSLS at h = 908, and viscosity, and temperature (in C). Up to about 500 s pure water flowed through the detectors, after which 5% was pulled from the reactor and the other 95% from the pure water reservoir, for the rest of the experiment. The flow rate through the detector train was ml/min throughout the experiment. The reactor feed flow rate r, was 3.13 ml/min up until about t = 7400 s. The RI and UV detectors reached their plateaus due to the 5% monomer flow by about t = 400 s, at which point the reactor temperature was increased from T = 58C to T = 608C within 110 s. The abrupt temperature rise in the reactor is clearly seen in Figure a. The high temperature caused the persulfate initiator decomposition to begin and hence started the polymerization reaction. The exponential drop of the UV marks the approach to the reactor steady state at r = 3.13 ml/min. The plateau reached by the UV indicates a steady concentration of monomer in the reactor. The increase and plateaus in TDSLS and viscosity signals show the appearance of Figure. (a) Raw data for experiment #1 (Table 1) in which the flow rate was ramped from high to low values. The numbers over the viscometer signal indicate reactor feed rate r, in ml/ min. The initiator concentration was 1.6 mg/ml, yielding [m] 0 / [I ] 0 = 79. (M/M). (b) Raw data for experiment #. The initiator concentration was 0.8 mg/ml, yielding [m] 0 /[I ] 0 = 159 (M/M). The detector voltages are in arbitrarily scaled units so that all can be shown simultaneously. The numbers over the UV data indicate reactor feed rate r in ml/min. polymer in the reactor. T remains at 59.3 8C l 0.5 8C throughout the reaction. At t = 7400 s, r was decreased to.66 ml/min, and, again, the exponential approach of the UV to the new

Online Polymerization Monitoring in a Continuous Reactor 591 Figure 3. Monomer conversion f, and M w, vs. t for the data from experiment # (Figure b), where M w was computed according to Equation (19). steady state is seen. Five more changes in r were then made; to r = 0..18 ml/min at t = 1000 s, to 1.71 ml/min at t = 16900 s, to 1.6 ml/min at t = 1000 s, to 0.83 ml/ min at 4600 s, and to 0. ml/min at 8900 s. The relative trends in the TDSLS and viscometer data are more complex than the UV data, which falls monotonically with decreasing r, since conversion is higher (Equation (16)). Both the TDSLS and viscometer reach maximum values then decrease as r decreases further and conversion increases further. Since M w from TDSLS is proportional to the ratio of intensity/concentration to first order (the A c correction term in Equation (19) is used in the actual M w computations, as mentioned), the decrease in this ratio indicates that M w is decreasing with increasing conversion, a consequence of the QSSA, seen in Equation (6). The same argument holds true for the viscometer; i.e. g r, which is proportional to about M 0.75, is proportional to the viscometer signal divided by c, again indicating, independently, that the mass is decreasing with conversion. Figure b shows similar raw data for experiment #, whose conditions are summarized in Table 1. Figure 3 shows monomer conversion f, and M w, vs. t for the data from experiment #, where M w was computed according to Equation (19). Figure 4 shows M w and g r vs. f for experiment #1. Because f approaches its steady state value at a different rate than M w and g r, as discussed below, there is a phase lag between these latter values and f, so that accurate kinetic rate data cannot be taken directly from these curves, although they will serve as a rough approximation. Rather, it can be seen that there is a higher density of points for both M w and g r when the plateau in f is reached. The higher density of points is apparent in Figure 4 because only every 0 th point has been plotted for both M w and g r. These values can be used for kinetic rate determination. Figure 4. M w and g r vs. f for experiment #1. The points for each steady state conversion plateau are taken from the bottom of the dense collection of points seen at intervals. Every 0 th data point was plotted in order to highlight the high point density from the steady state plateau seen in Figure a. (b) Figure 5. (a) M w, inst taken from the plateau values of Figure 4 for experiment #1 are shown vs. f. The intercept is 6.8610 5, and the slope is 6.30610 5, in good agreement with Equation (). Similar M w,inst data for experiment # are also shown. Additionally, [g] 1.33 (f = 0) [g] 1.33 (f) for experiment #1 is shown, whose linear dependence on f is predicted by Equation (6). (b) log(m w) vs. log([g]), with the corresponding power law.

59 B. Grassl, W. F. Reed Figure 5a shows the plateau values of M w vs. f, for experiments #1 and #, along with the linear fits. In the QSSA, Equation (5) and (7) can be combined to yield M n, inst = M n, 0 (1 f) () In other words, the instantaneous value of M n falls linearly to zero with increasing conversion. On the steady state plateau, when M w ceases to change, the detectors are reporting the instantaneous value of M w (M w, inst) in the reactor at conversion f, which does not change as long as the steady state is maintained. Hence, Figure 5 a is equivalent to M w, inst(f) vs. f. Instantaneously, M w = M n for termination by disproportionation, so that M w, inst obeys an identical equation to Equation (), except with M w,0 in place of M n, 0. Equation () predicts that the ratio of the slope in Figure 4 to the y-intercept should be 1. In fact, the ratios from Figure 4 for experiments #1 and # are 0.93 and 0.84, which is in good agreement with Equation (). Table 1 summarizes the fits for these data. Figure 5a also shows [g](f = 0) 1.33 [g](f) 1.33 for experiment #1. g r also reaches its instantaneous value at the steady state. The relation between intrinsic viscosity and M can normally be expressed by a power law relationship of the form (Mark-Houwink equation) ½gŠ w ¼ km b w ð3þ where the mass average quantities have been used, since these are the ones measured by both the light scattering detector and the viscometer. The viscometer, in conjunction with the RI, allows reduced viscosity to be directly measured, via the definition g r ðf Þ ¼ gðf Þ g s cg s ¼ ð½gšðf Þ þ j½gšðf Þ cðf ÞÞ ð4þ where [g](f) is the intrinsic viscosity of the polymer at conversion f, and j is a constant equal to about 0.35 for neutral polymers. [6] The values of [g] are on the order of 500 cm 3 /g or less, and c(f) is less than 0.001 g/cm 3 for all values of conversion, so that the second term on the right of Equation (4) is small. Using j = 0.35, and solving Equation (4) for [g], shows that the maximum correction to g r to obtain [g], is 5%. The inset in Figure 5b shows the log(m w) vs. log([g]) plot, with the parameters k = 0.015 and b = 0.75, with [g] expressed in cm 3 /g. While we do not expect these parameters to necessarily hold over a wide range of M, since the log-log fit spans less than an order 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 conversion plateaus should hence depend on M w,inst(f) according to ½gŠ w;inst ðf Þ ¼ kðm w;inst ðf ÞÞ b which, in turn, can be expressed in terms of f as ð5þ Figure 6. p/(1 f) vs. p, according to Equation (7) for experiments #1 and #. The incept yields the polymerization rate constant a = k p [R] for each experiment. The intercept for experiment #1 is 7.4610 4 s.1, and for experiment # it is 5.0610 4 s 1. ½gŠ 1:33 ðf ¼ 0Þ ½gŠ 1:33 ðf Þ ¼ ð0:015þ 1:33 k p ½mŠ 0 k t ½RŠ f ð6þ This latter function is also plotted in Figure 5a, where the predicted linear dependence is seen to hold. Referring to Equation (16), the quantity p/(1 f) for the steady state values of f should yield a straight line, given by p ¼ a þ p ð7þ 1 f Figure 6 shows this representation for experiments #1 and #. The intercept yields the first order rate constant a = k p R = 7.4610 4 s 1, and 5.0610 4 s 1, for experiments #1 and #, respectively. Varying Initiator Concentration at Constant Reactor Feed/Withdrawal Rate Since the foregoing analysis has shown that it is possible to measure both k p [R] and M w as reactor conditions change, it became evident that a more complete study of reaction kinetics, in terms of the relation between Fk d, M w, k p /k t and [I ] 0, could be made in a single experiment by changing initiator concentration at intervals. This represents a concise, economical means of determining reaction kinetics without the need to run several independent experiments. Figure 7 shows M w and f vs. t for experiment #3, in which p was kept constant (7.13610 4 s 1 ), and the amount of initiator was increased at intervals. This was accomplished by switching the peristaltic feed tube into a fresh reservoir containing the same concentration of AAm (0.034 g/cm 3 ) and the desired new concentration of initiator. Simultaneously, initiator was also injected into

Online Polymerization Monitoring in a Continuous Reactor 593 Figure 7. M w and f vs. t for experiment #3, in which p was kept constant and the amount of initiator was increased at intervals. The numbers over the M w data are the initial initiator concentration in 10 3 m/l. the reactor to bring it up to the level of the new reservoir, in an effort to minimize the time of approach to the new steady state. Actually, this had the effect of transiently disturbing the plateau. The initiator increase at t = 6000 s was made only by switching to a new AAm/initiator reservoir without changing the reactor s initiator concentration. This eliminated the transient disturbance in the conversion curve. From Equation (4) and (7) it is easily shown that the cluster of kinetic constants can be determined by plotting! k t f ½I ;0 Š ¼ p ð8aþ Fk D kp 1 f where f a ¼ p ð8bþ 1 f Figure 8 shows this representation, from which the slope yields! k t ðm N s =LÞ ¼ 9075 ð9þ Fk d kp Since the behavior of both the approach to equilibrium and the equilibrium plateau itself confirm that the reactor is perfectly mixed, for all practical purposes, and that the polymerization proceeds according to the QSSA, it is possible to exploit the M w data in Figure 7 to complete the determination of the kinetic parameters. The kinetic parameter k p /k t can be computed from the above slope, together with the relation dm w ðf ¼ 0Þ k p ½mŠ sffiffiffiffiffiffiffiffiffiffiffi ¼ M AAm p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 Fk d k t = d ½I Š 0 ð30þ Figure 8. [I ] 0 vs. [pf/(1 f)] according to Equation (8) for experiment #3, including the corresponding linear fit and parameters. Figure 9. M w(f = 0), from extrapolation according to M w(f)/(1 f), vs. 1/sqrt([I ] 0 ) for experiment #3. Also shown are the M w(f = 0) values for experiments #1 and #, as well as M w(f = 0) data taken from Table 1 of ref. [14] Slope to experiment #3 is 74800. Circles represent experiment #3, squares are from Table 1 of ref., [14] and diamonds are experiments #1 and #. with M AAm = 71 g/mole, and [m] 0 = 0.479 M/L, via k p k t ¼ 1 dm w ðf ¼ 0Þ sffiffiffiffiffiffiffiffiffiffiffi ½mŠ 0 M AAm 1 d ½I Š 0 1= ð31þ d½i Š 0 da Figure 9 shows M w(f = 0), from extrapolation according to M w(f)/(1 f), vs. 1/sqrt([I ] 0 ). The value of k p /k t is 11.7 L/M N s. Also shown in Figure 9 are the M w(f = 0) points for experiments #1 and #, as well as the data for M w(f = 0) taken from Table 1 in reference 14. All the experiments

594 B. Grassl, W. F. Reed are in good agreement, except for one point (M w(f = 0) L 610 6 ) that deviates significantly from the other data. Kinetics during the Approach to Steady State Equation (15) (17) above indicate how c t (t), c m (t) and c(t) vary as a function of time for different combinations of flow, polymerization and concentration conditions. Figure 10 shows an excerpt from Figure 7 (experiment #3) of one of the exponential approaches to the steady state. The exponential fit is quite good, in agreement with Equation (9), which confirms two important assumptions made: First, the reaction indeed proceeds via the QSSA, and, second, the mixing of the reactor is perfect, for all practical purposes. This allows the rate constant to be determined alternatively via the rate constant of the exponential approach to equilibrium. An analysis of the approach of M w(t) to the steady state requires a theoretical expression. At any time t, the concentration of dead polymer chains in the reactor will be the sum of the remaining concentration increments of polymer that were produced ever since t = 0. As noted above, the time it takes to produce a polymer chain is far shorter than 1/p, the average residence time in the reactor. M w(t) is, by definition M w ðtþ ¼ Z t 0 M w;inst ðt9þdcðt; t9þ Z t 0 dcðt; t9þ ð3þ M w, inst(t9) is the instantaneous value of M w(t), and can be found by Equation (7) above as k p c m ðtþ M w;inst ðtþ ¼ k t ½RŠM AAm ð33þ where the factor of is the ratio of the instantaneous mass and number averages for free radical reactions terminating by disproportionation (if recombination reactions are also involved the factor would be less than ), and mass concentration units are used for monomer. Now dc(t,t9) is the increment of concentration of polymer of mass M w, inst(t9), produced at time t9, and still left in the reactor at time t. In other words, it is the product of the amount of polymer produced at t9, times the fraction of polymer remaining at t, dcðt; t9þ ¼ 4 qc qt t9;due to polymerization 3 5x½A exp½ pðt t9þššdt9 ð34þ Here, Aexp[ p(t t9)] is the probability that a mass element that entered the reactor at time t9 is still left in the reactor at time t, assuming the reactor is perfectly mixed, as defined above. A is a normalization constant that ensures cðtþ ¼ Z t 0 Z t dcðt; t9þ ¼ AaM AAm c m ðt9þexp½ pðt t9þšdt9 ð35þ and the fact has been used that the amount of polymer produced between the interval t9 and t9+dt9 is 4 qc qt q½mš qt t9;due to polymerization t9; due to polymerization 0 3 5 ¼ M AAm ¼ M AAm ac m ðt9þ ð36þ Integrating Equation (35), and using Equation (16) for c(t) allows A(t) to be determined, AðtÞ ¼ ac m;s þ ðc m;r c m;s Þexpð ptþ c m;r pc ;m;s exp½ ðþtš c m;s a expð ptþ ½expðptÞ 1Š þ 1 c m;r pc ð37þ m;s ½1 expð atþš a Using this in conjunction with Equation (35) for M w, inst (t9) yields M w ðtþ ¼ 8 >< p >: c m;s ½expðptÞ 1Š þ pc m;s aðþ k t ½RŠM AAm k p c m;r pc c m;r pc 9 m;s >= m;s ½1 expð atþš þ ½1 exp½ ðp þ aþtšš >; ð38þ c m;s ½expðptÞ 1Š þ 1 a c m;r pc m;s The case most used in this work is that c m, r = c m, s. Then M w (t) simplifies to M w ðtþ ¼ k p c m;s k t ½RŠðÞ ½1 expð atþš p½expðptþ 1Š þ p½1 expð atþš þ a ½1 exp½ ðp þ aþtšš p þ a expðptþ expð atþ ð39þ

Online Polymerization Monitoring in a Continuous Reactor 595 i.e. the effective rate constant of the M w(t) approach to equilibrium is approximately one half the rate constant of f(t) approach to equilibrium. Figure 10 also shows M w(t), together with f(t), and it is seen that this relationship is closely born out. It is also generally found in the other experiments for each individual parameter change leading to a new steady state. The approach of g r to the steady state can be found by invoking the scaling law in Equation (3). Then Figure 10. f and M w vs. t for one of the exponential approaches to the steady state for experiment #3. M w approaches the steady state in a pseudo-exponential fashion, in which the effective rate constant is about half that of the rate constant governing the approach of f to the steady state, according to Equation (4). The inset shows the pseudo-exponential approach of g r to the steady state, at a rate less than M w(t), as predicted by Equation (43). This latter expression implies that the approach to equilibrium of M w(t) will always be slower than for f(t), as is now demonstrated. Although M w(t) is not a pure exponential decay function, the time dependent portion resembles one, in which the initial value is 1 and the final value is p/(p + a), as can be readily verified by taking the limits of t = 0 and large t in Equation (39). Hence the amplitude of the time dependent portion is a/(a + p). Now, if the limit of small t is taken of Equation (39), by expanding the exponentials in the numerator and denominator to second order in t, using the binomial expansion on the denominator, and retaining terms to order t, one obtains, after considerable algebra M w ðtþ L k pc m;s k t ½RŠ 1 at ð40þ Equating the time dependent part of this to the first order expansion of an effective exponential that has the prefactor a/(a + p) and a decay rate b, M w ðtþ lim M w ðtþ ¼ k pc m;s tev k t ½RŠ L k pc m;s k t ½RŠ a ð1 btþ 1 at p ð41þ yields an important approximate relationship between the decay rate of the effective decay of the time dependent part of M w(t) and the constants a and p, namely b X a þ p ð4þ ½gŠ w ðtþ ¼ k Z t 0 M w;inst ðt9þ b dcðt; t9þ cðtþ ð43þ Since b in general can range from 0.5 to 0.8 for coil polymers, no single expression can be given for the above integral. However, since it represents an average proportional to pm b P, we can surmise that the decay rate will be lower than for M w(t). This qualitative observation is born out in the inset of Figure 10, where g r (t) is plotted f, and the pseudo-exponential decay rate is found to be 73% of the M w(t) pseudo-exponential decay rate. Although we do not wish to pursue the details here, for completeness it is mentioned that the polydispersity index M w(t)/m n(t) during the approach to the steady state can be computed by finding M n(t9) using the above method. The number average polymer mass M n(t) is M n ðtþ ¼ ¼ Z t 0 cðtþ dcðt; t9þ M n;inst ðt9þ pk p cðtþ aaðtþexpð ptþk t ½RŠM AAm ðexpðptþ 1Þ ð44þ where the QSSA form for M n, inst (t9) = M AAm N n, inst from Equation (7) has been used. Approximation to a Fully Continuous Reactor Equations (16) and (38) imply that if the initial monomer concentration in the HCSTR is less than that in the reservoir, and obeys the relationship c m;r ¼ pc m;s ð45þ then c m (t) and M w(t) will be immediately in the steady state as soon as the reaction begins, and the polymer concentration will exponentially approach its full steady state value with the rate constant p. This would then allow the HCSTR to approximate a fully continuous tube reactor in terms of c m (t) and M w(t). Figure 11 shows c m (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

596 B. Grassl, W. F. Reed Figure 11. M w(t), c m (t) and c p (t) for experiment #4, which attempts to approximate a fully continuous reactor. Figure 1. M w(f) for a batch reactor. The inset shows f(t), along with a first order fit. perfect approximation was not achieved, due to error bars in both the experimental value of a, and in practically establishing the reaction concentration and other conditions, it is clearly seen how greatly diminished the amplitudes of c m (t) and M w(t) are in their approach to the steady state. The large amplitude change in c m (t) that would occur is seen from the data, where the system was first pumped at the full reservoir monomer concentration of c s = 0.034 g/cm 3, before switching to, and stabilizing at, the initial reactor concentration of c r = 0.013 g/cm 3. Although the amplitudes are greatly diminished, the rate constants at which these residual deviations from the steady state value relax involve the same rate constants as when no attempt at such a balance is made. As expected the initial values of M w(t) are smaller than at the steady state, because c m,r a c m,s and the amount of initiator is constant in both the reactor and reservoir at all time. This is predicted by Equation (34), due to negative prefactors in two of the terms. Comparison with Batch Reactor Results To contrast the HCSTR results, Figure 1 shows the results for M w(f) for a batch reactor. The AAm initial concentration was, 0.034 g/cm 3 as in most of the other experiments, and [I ] 0 /[AAm] 0 = 0.094 g/g. The inset to Figure 1 shows f vs. t. A first order fit is also shown, but is so close to the data that it is difficult to discern. The first order nature of f(t) is evidence that the QSSA is approximated by this batch reaction. The linear decrease in M w over most of the conversion is also characteristic of the QSSA. The batch reactor measures the cumulative weight average mass of all dead polymer chains M w, cumulative, which, in the QSSA is given by M w; cumulative ðf Þ ¼ M w;0 1 f ð46þ Figure 13. An HCSTR reaction in which temperature, initiator concentration and mixing were allowed to fluctuate. This is experiment #5 in Table 1. This approximation is closely obeyed in the figure for conversion values above 0.. Deviations of the PAAm reaction from ideal polymerization were discussed in reference 14. Monitoring Fluctuations in Conversion Due to Drift in Reactor Conditions Whereas the previous experiments have shown how steady state HCSTR operation is approached and can be maintained, there is considerable practical interest in being able to monitor the effects of drift in reactor conditions. Figure 13 shows the results of experiment #5, in which a steady state was first achieved, then the following manipulations made: 1) deliberate temperature fluctuations, ) the effect of a change in initiator concentration, and 3) non-perfect mixing.

Online Polymerization Monitoring in a Continuous Reactor 597 The most dramatic effects on the reaction are seen when the stirring rate is changed. During the last portion of the curve, stirring speed was reduced to nearly zero, then to zero altogether. The low conversion in this portion, and corresponding high M w (which follows from a high monomer to initiator ratio at low conversion), may be the result of monomer passing quickly from the nearly static reactor solution directly to the withdrawal tube, leaving little time for conversion to occur. This process is chaotic, leading to random fluctuations in the conversion. Conclusions The online polymerization monitoring method has been successfully adapted to a HCSTR. It has been demonstrated that conversion, and hence M w, can be tightly controlled by reactor feed/withdrawal rate. The exponential approach in time of f to steady state conditions immediately indicates that 1) the polymerization reaction is proceeding according to QSSA (first order) kinetics and ) that the reactor is well mixed. It has been further shown that the method provides a concise, economical method for determining detailed reaction kinetics, since a wide range of initiator/monomer ratios can be quantitatively monitored during a single experiment. It has also been demonstrated that, by proper choice of initial monomer concentration in the reactor and feed reservoir, the amplitude of the exponential approach to the steady state can be significantly reduced; i.e. the HCSTR can be made to approximate a continuous tube type reactor. Adaptation to continuous tube type reactors should be straightforward. Finally, the long term stability of the steady state was investigated, as well as the effects of deliberately causing conditions to drift. In this case, deliberate fluctuations in temperature, initiator concentration and stirring rate were made to show the deviations from steady state operation, those due to stirring being the most dramatic. [1] H. A. Dotson, R. Galvan, R. L. Laurence, M. Tirrell, Polymerization Process Modeling, VCH Publishers Inc., 1996. [] H. Tobita, J. Polym. Sci., Part B: Polym. Phys. 1994, 3, 911. [3] N. G. Podosenova, E. G. Zotikov, Soviet Chem. Ind. 1991, 3, 15. [4] G. W. Poehlein, Polym. Int. 1993, 30, 43. [5] D. M. Kim, B. E. Nauman, Ind. Eng. Chem. Res. 1997, 36, 1088. [6] M. A. Dube, J. B. P. Soares, A. Penlidis, A. E. Hamielec, Ind. Eng. Chem. Res. 1997, 36, 966. [7] I. Plazl, Ind. Eng. Chem. Res. 1998, 37, 99. [8] A. M. Ahn, M. J. Park, H. K. Rhee, Ind. Eng. Chem. Res. 1999, 38, 394. [9] F. Teymour, W. H. Ray, Chem. Eng. Sci. 199, 47, 411. [10] J. C. Verazaluce, A. F. Tlacuahuac, E. S. Guerra, Ind. Eng. Chem. Res. 000, 39, 197. [11] F. Tehmour, W. H. Ray, Chem. Eng. Sci. 199, 47, 4133. [1] F. H. Florenzano, R. Strelitzki, W. F. Reed, Macromolecules 1998, 31, 76. [13] W. F. Reed, Macromolecules 000, 33, 7165. [14] A. Giz, H. Giz, J. L. Brousseau, A. Alb, W. F. Reed, Macromolecules 001, 34, 1180. [15] B. Grassl, A. Alb, W. F. Reed, Macromol. Chem. Phys. 001, 0, 518. [16] A. Giz, H. Giz, J. L. Brousseau, A. Alb, W. F. Reed, J. Appl. Polym. Sci. 001, 8, 070. [17] P. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y. 1971. [18] J. L. Brousseau, H. Giz, W. F. Reed, J. Appl. Polym. Sci. 000, 77, 359. [19] W. M. Kulicke, R. Kniewske, J. Klein, Prog. Polym. Sci. 198, 8, 373. [0] R. Strelitzki, W. F. Reed, J. Appl. Polym. Sci. 1999, 73, 359. [1] D. P. Norwood, W. F. Reed, Int. J. Polym. Anal. Charact. 1997, 4, 99. [] J. P. Riggs, F. Rodriguez, J. Polym. Sci., Polym Chem. Ed. 1967, 5, 3151. [3] D. Hunkeler, Macromolecules 1991, 4, 160. [4] A. S. Sarac, Prog. Polym. Sci. 1999, 4, 1149. [5] D. Hunkeler, A. E. Hamielec, in: Water Soluble Polymers, S. Shalaby, C. L. McCormick, G. B. Butler, Eds., Am. Chem. Soc., Washington D.C. 1991. [6] M. L. Huggins, J. Am. Chem. Soc. 194, 64, 1093. Acknowledgement: The authors gratefully acknowledge support from the U.S. National Science Foundation CTS 987706. Received: August 9, 001 Revised: October 10, 001 Accepted: October, 001