a sensitivity approach to reachability analysis for particle size distribution in semibatch emulsion...

A Sensitivity Approach to Reachability Analysis for Particle SizeDistribution in Semibatch Emulsion Polymerization

Charles David Immanuel and Francis Joseph Doyle, III*

Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

Experimental and simulation-based sensitivity studies on the process of evolution of particlesize distribution (PSD) in semibatch emulsion polymerization are presented. The experimentalstudies identify an effective control strategy for PSD, and the appropriate manipulated variables.The complementary simulation studies identify the potentially reachable PSDs in the complexoperating space. The study also examines the effect of uncertainties and disturbances on thereachable distributions, to evaluate the benefits of in-batch feedback control.

1. Introduction

The control of distributed chemical processes is gain-ing increased attention due to the potential process andproduct improvement that is obtained through directoptimization of quality variables.1-4 For example, theparticle size distribution (PSD) in emulsion polymeri-zation is critical in determining the end productsproperties of adhesion, rheological properties, and me-chanical strength. The target distribution in most casesis multimodal and non-Gaussian. The control of distri-butions and profiles motivates a detailed analysis of thecontrollability of these systems. The mechanism offormation of distributions and profiles are intricate andare influenced in complex and in some cases nonintui-tive ways by the inputs available in the process, hencethe need for the sensitivity analysis, to evaluate thefeasibility of control of distributions and identify aneffective control strategy and a suitable combination ofmanipulated variables. In addition to these sensitivitystudies, a very pertinent issue is the identification ofclasses of distributions that can be produced in theseprocesses. In the highly multivariate character thatthese distributions represent, the attainable regions ofthe different variables could be correlated, therebylimiting the attainable classes of distributions. Whileit is easier to ascertain these controllability issues forlinear lumped parameter systems, it is not so in the caseof the nonlinear distributed parameter systems. Thispaper first presents experimental sensitivity studies onthe process, to demonstrate the feasibility of control ofdistributions. The second objective of the paper is toanalyze the reachability of the system, namely, theability to steer an output from an initial state to a finalstate in a finite time, using the available inputs.Controllability includes the ability to maintain theoutput at this final state beyond the end point. Thereachability problem is approached by employing adetailed population balance model of the system.

The focus of the present study is on emulsion polym-erization, in which the particle size distribution (PSD)is determined by the three major phenomena of nucle-ation, growth, and interparticle coagulation. One has a

variety of potential inputs to be employed for the controlof PSD. These include the feeds of the surfactants(emulsifiers), monomers, and initiators, the concentra-tion of the reagents, and the temperature. It is impor-tant to identify the best inputs and the best controlconfiguration. While most of the (few) studies on thecontrol of PSD have focused on surfactant alone, orsurfactant and initiator as manipulated variables,Crowley et al.5 examined the effectiveness of controlwith surfactant feed directly as a manipulated variable,and also with surfactant concentration in the aqueousphase as a manipulated variable. Meadows et al.6examined the suitability of temperature as a manipu-lated variable for PSD control. Liotta et al.7 consideredthe weight fraction polymer within the particles as themanipulated variable, which in turn was regulatedusing monomer feed, for the control of diameter ratioof a seeded bidisperse population (lumped variable).

Distributed parameter systems (DPS) are character-ized by partial differential equations, in the case underconsideration the population balance equation, resultingin infinite-dimensional systems. These systems can beapproximated as finite-dimensional multivariable sys-tems using suitable discretization techniques, with avery high ratio of the number of correlated outputs(controlled variables) to the number of inputs (manipu-lated variables). As stated above, the correlation andinteraction between the nucleation, growth, and coagu-lation events sets certain constraints on the types ofdistributions that can be produced in the emulsion. Evenin the case of a monodisperse population, there is anupper limit on the particle size to yield a dispersion ofsolids in the aqueous phase with an appreciable solidscontent in the latex. As the particle size of the mono-disperse population increases, the achievable solidscontent is reduced. Further, the relative rates of nucle-ation and growth limits the attainable polydispersityof monodisperse populations. Allowing for a distributionof particle sizes in the population expands the attainabledomain of solids content. However, the attainabledistributions are dictated by the achievable rates ofparticle nucleation and growth and by the particlestability, in addition to the strong interaction amongthese. Although there are process inputs to manipulatesome of these subprocesses independently, there is alsoa high degree of coupling with certain inputs. Consider-ing all these issues and the constraints on the inputs,

* To whom correspondence should be addressed. Presentaddress: Department of Chemical Engineering, University ofCalifornia at Santa Barbara, Santa Barbara, CA 93106. Tel.:(805) 893-8133. Fax: (805) 893-4731. E-mail: [email protected].

327Ind. Eng. Chem. Res. 2004, 43, 327-339

10.1021/ie030145p CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 08/19/2003

the actual attainable distributionssboth unimodal andmultimodalsare limited.

Formal controllability studies on particulate systems,aimed at determining the controllable distributions, arerather limited. To avoid unattainable set points, somestudies in the past on the control of distributions haveutilized a partial control-like strategy in which a few ofthe outputs are controlled while the rest are allowed toevolve in an open-loop manner.8,3 Semino and Ray9

addressed the issue of controllability of various particu-late systems including emulsion polymerization. Theyaddressed the control of PSD in emulsion polymeriza-tion by manipulating the nucleation phenomenon. Con-sidering that the micellar nucleation phenomenon isinfluenced by the feeds of the surfactant and that of theinitiator and/or the inhibitor, they found that thenucleation phenomenon can be completely controlledwhen the feeds of either the surfactant or the initiator/inhibitor is unconstrained. A strategy for determiningthe attainable bounds was developed by Glasser et al.,10

and demonstrated by Smith and Malone,11 for moresimple problems. Liotta et al.7 employed a similarstrategy in their study on the reachability of diameterratio (lumped) of bidisperse populations by manipulat-ing the competitive growth phenomenon in seededemulsion polymerization of styrene. They identified thereachable regions of diameter ratio and examined itssensitivity to the initial conditions. The upper and thepractical lower limits on the monomer concentrationwithin the particles were used to identify the reachableregions. Other studies employ a similar philosophy, butrely on simple simulations in determining the reachableregions and in analyzing the processes.12

Recently, there have been studies that address thedevelopment of the reachable regions through thesolution of optimization problems.13,14 The latter studydeals with determining the reachable regions of PSDin styrene emulsion polymerization. In this study, Wangand Doyle III have identified the reachable domains interms of the “ε-reachability”, defined to be the domainof distributions that are reached subject to a toleranceε in the defined objective function.

In the current study, a combination of experimentsand simulations using population balance models areemployed to analyze the evolution of PSD in thesemibatch emulsion copolymerization of vinyl acetate(VAc) and butyl acrylate (BuA), using nonionic surfac-tants. The experimental studies are mainly used toidentify a suitable control configuration and the ma-nipulated variables. The analysis addresses the hier-archy of the individual subprocesses of nucleation,growth, and coagulation, where possible. The simulationstudies extend the analysis to identify the reachablebimodal distributions, with the objective of extractinginformation on the capabilities and limitations of theprocess with regard to the control of distributions.

2. Experimental Sensitivity Studies

A state-of-the-art experimental facility was utilizedto study the sensitivities in the process of evolution ofPSD. The facility includes a 3-L stirred reactor, equippedfor semibatch operation and provided with instrumen-tation for on-line measurement of latex density andPSD. See ref 15 for a detailed description of the facilityand for the reproducibility characteristics of the equip-ment and the overall process. The system under inves-tigation is vinyl acetate (VAc)-butyl acrylate (BuA)

emulsion copolymerization. A nonionic surfactant isused as the emulsifier, with a redox initiation mecha-nism (tert-butyl hydrogen peroxide, t-BHP, and sodiumformaldehyde sulfoxylate, SFS).

In interpreting the experimental results, it is impor-tant to understand the underlying mechanisms of theevolution of PSD. In emulsion polymerization, thepolymer is produced as a colloidal dispersion of particlesin the aqueous phase. Although the polymerization isprimarily initiated in the aqueous phase (by water-soluble initiators), the predominant locus of polymeri-zation lies within the particles. Thus, each of theseparticles constitutes a microscale bulk polymerizationreactor, comprising active and terminated polymerchains and the monomers. The monomers are usuallysparingly solubilized in the aqueous phase and areusually in a thermodynamic equilibrium between theaqueous phase and the particles. When these phases aresaturated with the monomers, separate monomer drop-lets result as additional dispersed phases. The dispersedphases are stabilized by surfactants, which, by virtueof their amphiphilic character, adsorb onto the dispersedphases, and thereby keep them apart in the aqueousphase.

The PSD evolves according to the interplay betweenthe phenomena of particle nucleation, growth, andcoagulation. When the concentration of surfactant in theaqueous phase exceeds the critical micelle concentrationvalue of the surfactant, the surfactants form micellesthat serve as nucleii for particles. Particles are formedby the entry of a polymer radical from the aqueousphase into the micelles. Particle nucleation can alsooccur under micelle-free conditions by the precipitationof a polymer radical initiated in the aqueous phase uponthe attainment of a critical chain length, correspondingto the solubility of the radicals in water. The nucleatedparticles grow by the polymerization of the radicalswithin them with the monomers absorbed into theparticles. The particles can also grow in discrete leapsby coagulating with each other, which occurs due to theinstability of the particles. The nonionic surfactantsemployed in this study have the tendency to partitioninto the dispersed phases, unlike the ionic surfactantswhich restrict themselves to the aqueous phase and theinterfaces. This aspect results in complications mainlyin the nucleation pattern, as was demonstrated previ-ously.16,17,15 Thus, the identification of the best controlinputs assumes greater intricacy while employing non-ionic surfactants. The series of experiments aimed atanalyzing the process sensitivities are described next.

Figure 1a shows the base case feed profiles pertainingto the two monomers VAc and BuA (pure components),surfactant solution (22.7 wt % in water), t-BHP solution(3.33 wt % in water), and SFS solution (3.41 wt % inwater). The initial mixture consists of 1-L of DI water,52 g of VAc monomer, and 0.1 g of ferrous ammoniumsulfate for the coordination of the redox initiation.Figure 1b shows the evolution of the PSD during thecourse of the batch, and Figure 1c shows the corre-sponding profile of total particles. Figure 1c indicates agradual and prolonged nucleation event, which coupledwith the size-dependent growth phenomenon and par-ticle coagulation results in a broad and interconnecteddistribution at the end of the batch (Figure 1b). Figure1d shows the profile of solids content, as obtained fromgravimetry and from the densimetric calculations pre-

328 Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004

sented in ref 15. It records a modest 21% solids contentat the end of the batch.

This base case recipe is subjected to various perturba-tions, and their effects on the process are studied. Thesestudies are used to infer the ability to steer the entiredistribution in certain directions and to identify effectivecontrol configurations. The experiments and the infer-ences from them are described below.

2.1. Effects of Surfactant. The experiment de-scribed in this subsection was designed to determine theeffect of changing the surfactant feed profile on the PSDand the conversion of the monomers. The surfactantfeed rate was perturbed as shown in Figure 2a. Figure2c compares the profiles of total particles in the currentand the base case experiments. The total number ofparticles nucleated in this experiment is much lower(∼1016), and also nucleation is not prolonged, unlike inthe previous experiment. This is mainly due to thedecrease in the surfactant feed rate between 10 and 15min (compared to the base case experiment). Thedecrease in the number of particles in the latex resultsin larger growth rates, thereby causing the particles togrow to a larger size (Figures 2b). This figure also showssome large size particles (larger than 600 nm), suggest-ing a coagulation event that could have occurred duringthe reduced surfactant feed rate. The end-point solidscontent increases to about 25% at a final overallconversion of about 95% (almost 20% higher than in thebase case). This aspect clearly demonstrates the complex

and nonintuitive behavior of the system. Increasing thesurfactant feed rate to effect secondary nucleation anda bimodal end-point distribution could cascade intolower growth rates and particle sizes, and hence a lowerconversion. This necessitates using other inputs in theprocess, such as the feed rates of initiators and mono-mers, to correct this situation. Another aspect of theprocess that is evident from this experiment is thecompetitive particle growth phenomenon. This is seenin Figure 2d, which depicts the evolution of the bimodaldistribution (number-averaged plot) along the course ofthe batch. It shows a more interconnected and diffusedistribution at the intermediate time range, whichbecomes separated into a more clear bimodal distribu-tion toward the end of the batch. This is due to thestrongly size-dependent growth, with the larger par-ticles growing faster than the smaller ones. The size-dependent growth and the lack of inputs to manipulatethis phenomenon demonstrate the importance of thetimings and durations of the various nucleation events.In producing a multimodal distribution, the polydisper-sity of each mode and the separation between the modesis mainly influenced by the duration and the timings ofthe nucleation events.

2.2. Effects of MonomersVinyl Acetate. The nextexperiment was designed to investigate the influenceof the feed rate of one of the monomers on the evolutionof the PSD and other pertinent outputs. Vinyl acetatefeed rate was perturbed as shown in Figure 3a. Figure

Figure 1. Results corresponding to the base case ab initio experiment.

Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 329

3c,d compares the profiles of total particles and solidscontent between the two experiments. Despite thereduced feed rate of the VAc monomer at the start ofthe reaction, there is no appreciable difference betweenthe two cases at early times (through 30 min) of thebatches. A plausible explanation is that the system stillremains saturated with the monomers. But unlike inthe base case, there is a steep increase in the numberof particles at approximately 37.5 min, coinciding withthe drop in the VAc feed rate at this time (Figure 3a).One plausible explanation is that all the monomerdroplets disappear rapidly from the system, coincidingwith the decrease in the feed of VAc at 37.5 min. Thisresults in releasing all the absorbed surfactants backinto the aqueous phase and causing rapid nucleation.The end-point distribution (Figure 3b) shows appre-ciable mass of very small particles, due to the largernumber of particles nucleated at the later times and alsodue to the prolonged monomer addition in this experi-ment. The solids content in this experiment increasesby about 3%. These results clearly show that themonomer feed affects the PSD by influencing not onlythe rate of growth but also the rate of nucleation. Whilea monomer-starved condition might decouple the effectof monomer feed on the nucleation process, it wouldresult in reduced growth rates.

2.3. Effects of MonomersButyl Acrylate. Figure4 depicts results corresponding to a perturbation in thefeed rate of BuA monomer (shown in Figure 4a). Thereis a much reduced nucleation rate in this case, as is seenfrom the profile of the total particles (Figure 4c), andthe nucleation process continues through the course ofthe batch. The end-point PSD (Figure 4b) shows asignature of coagulation in the form of the largeparticles (larger than 600 nm). The results suggest acomplex dependence of the surfactant partitioning onthe monomer composition. One explanation is that thesurfactant solubility in the monomer droplets is largerat higher VAc composition in the droplets.

2.4. Effects of Initiator. Figure 5 shows sensitivityresults from an experiment in which the concentrationof the redox initiator pair (t-BHP and SFS) were doubledrelative to the base case, while the feed rates weremaintained the same as in Figure 1a. Figure 5bcompares the profiles of total particles, which shows thatin this case there is a larger nucleation event at theinitial times (up to 15-20 min). Thereafter, the totalnumber of particles remains relatively constant untilapproximately 40 min, at which time there is a secondnucleation event. Thus, the end-point distribution is amore prominent bimodal distribution (as seen in Figure5a). This could be attributed to a larger nucleation eventat the early times (by either of micellar or homogeneous

Figure 2. Effect of a perturbation in the surfactant feed rate relative to the base case experiment.


mechanisms or both), which depletes the micelles andthereby prevents further nucleation events until ap-proximately 40 min when more micelles are formed. Thenew batch of micelles could be due to either the increasein the surfactant feed or the drop in the VAc feed (andthe associated depletion of the droplets). Thus, there aretwo clearly distinct nucleation events, contrary to theprolonged micellar nucleation event that characterizesthe base case experiment. This observation clearlydemonstrates the strong ability to influence the nucle-ation events through initiator feed rates or composition.Thus, the rate-limiting step in this case is clearly theformation of micelles. However, there is no appreciableeffect on growth, as seen by the comparable particlesizes.

2.5. General Perturbation. A combined perturba-tion of the variables was performed to see its effect onthe process. Specifically, the feed of the surfactantsolution and the monomers were delayed relative to thatshown in Figure 1a, while the VAc monomer in theinitial mixture was left unchanged. This result waspresented in ref 15 in a related context. Its implicationunder the present context is the relatively strongsensitivity of the process to even small and inevitableprocess variations, such as the delays in the pumps, etc.and the inability to correct these errors using feedbackcontrol. This result highlights the irreversible natureof the process, and hence the need for very stringentoperating practice.

2.6. Implications of the Sensitivity Results forthe Control of PSD. In the experiments describedabove, the effects of the manipulative variables on thePSD and solids content were studied experimentally.Each of the feeds of the surfactant, monomers, andinitiators have profound influence on the evolution ofthe distribution. However, the observed influences arecomplex and nonintuitive. It is natural to expect thesurfactants to affect the nucleation phenomenon andparticle stability and the initiators and monomers toaffect mainly the growth phenomenon. Thus, it is logicalto utilize the feed rates of surfactant and initiator asmanipulated variables for the control of PSD, as wasdone with ionic surfactants by other researchers.5,18

However, in the current case, it is seen that all thereagentsssurfactant, monomer, and initiatorsinfluencethe nucleation phenomena, albeit in different ways.Monomer affects the growth phenomenon, while theeffect of the initiator on growth is seen to be minimal.Surfactants, whose major role is particle stabilization(emulsification), do affect the coagulation phenomenon.In addition to these direct effects, the variables alsohave secondary effects, which come into play becauseof the interaction among nucleation, growth, and co-agulation. For instance, the surfactants can affect thegrowth process indirectly, by affecting the number ofradicals/particle and the monomer concentration insidethe particles (by varying the particle number). Also, thebehavior observed is quite complex and cannot be

Figure 3. Effect of a perturbation in the VAc feed rate relative to the base case experiment.


adequately represented in terms of any simple dynam-ics, even for continuous processes (for example, firstorder, time delays, and inverse responses). Thus, adetailed model-based optimization and control strategyis appropriate for this process.

Another aspect that is revealed in the current studyis the strong size dependence seen in the growthphenomenon, which results in a relative broadening ofthe distribution with growth. Note that this is incontrast to the observations made by Liotta et al.,7 whoobserved a relative narrowing of seeded bidispersepopulations with growth. Modeling studies also supportthe observation on the broadening of the distribution.19

This is a clear illustration of the ineffectiveness oflumped approaches in certain cases and lends strongsupport to the distributed parameter route pursued byresearchers in recent years.4,3,1,2 This strong size de-pendence necessitates tight control on the timings,durations, and magnitudes of the nucleation events inproducing the desired distributions with particularmean sizes and standard deviations of the modes. Oncethe nucleation event has occurred, there is very littlelatitude to alter the competitive growth.

A third aspect that is evident from these experimentsis the irreversibility that is characteristic of theseprocesses. Each of the nucleation, growth, and coagula-tion processes exhibit a certain irreversible character.For example, when a unimodal distribution is beingproduced, if the actual nucleation rate deviated from

the desired rate, the effect of this error on the particlesizes can be corrected by suitably modifying the growthrate (by recruiting multiple process inputs). However,the effect of this error on the breadth of the distributioncannot be rectified (the import of this irreversibility onthe breadth of the distributions being dependent on theend applications). Similarly, in producing multimodaldistributions, if the nucleation rate for the first nucle-ation event is erroneous in implementation, this can becorrected (in a relative sense, and bearing with theskewness of the distributions) by correcting not only thegrowth but also all the subsequent nucleation rates. Onthe other hand, if the primary nucleation event isimplemented as planned, but the secondary nucleationevent is erroneously implemented, it might leave anuncorrectable effect on the distribution. Similarly, alarger growth rate might result in larger sizes, whichagain cannot be corrected (no shrinkage possible withrespect to the polymer mass in the particles). A strongcoagulation event would also leave an indelible markon the distribution (see ref 20 for examples of these).The interactive nature between nucleation, growth, andcoagulation again advises caution in the control ofdistributions.

From the preceding analysis, it is evident that onemust employ multiple inputs for the control of distribu-tions, with the surfactant and VAc monomer being themost suitable inputs. Initiator feed could also be re-cruited, but mainly to control the nucleation phenom-

Figure 4. Effect of a perturbation in the BuA feed rate relative to the base case experiment.


enon. A monomer-starved (droplet-free) condition mightbe preferable to render the nucleation event a preroga-tive of the surfactant feed alone (and the initiator feed,if utilized). However, too low concentrations wouldresult in suboptimal growth rates. Although coagulationcould potentially be used to shape the distributions, thebest strategy is to minimize coagulation events to theextent possible. Also, the irreversibility considerationsand the interactions suggest an hierarchical strategyin which the subprocesses (nucleation, growth, andcoagulation) are controlled individually, thereby produc-ing the target distribution. This hierarchical strategycan also be employed within a partial control configu-ration in which only a subset of these three major

subprocesses are controlled. Such a configuration wasactually employed in our control studies21 and is akinto the idea of partial control.22 A similar idea is alsoimplicit in the studies of Semino and Ray.9

3. Simulation-Based Reachability Analysis

In the previous section, experimental sensitivityresults were detailed, based on a perturbation in eitherthe timing of a step change in input or the magnitudeof a step change. A more exhaustive sensitivity studywould examine the effect of all possible such perturba-tions on the distribution. This analysis becomes almostimpossible experimentally (although the results pre-sented above are some of the most relevant perturba-tions). In addition to these open-loop studies, it is alsoof interest to study the effectiveness of feedback controlin eliminating the effects of uncertainties and distur-bances in the processsa robustness analysis for theopen-loop reachable distributions. A comprehensivepopulation-balance model has been developed for thisprocess, both without coagulation and incorporatingcoagulation events, and validated against experi-ments.19,23 It models the PSD in terms of a particledensity F(r,t) dr, which is defined as the moles ofparticles of size within a small interval r and r + dr.The PBE is given by

where the partial derivative with respect to r accountsfor the particle growth and Rcoag(r,t) accounts for thecoagulation events. Particle nucleation events, whichare restricted to the smallest particle size in thedistribution, enter through a boundary condition. Acomputationally efficient solution method has beendeveloped, which enables rapid simulations of themodel.24 This model can be utilized to perform morecomprehensive sensitivity studies. First, the open-loopanalysis is presented, followed by a robustness analysisto determine the effects of common disturbances anduncertainties.

3.1. Open-Loop Reachability. A typical semibatchrecipe is considered, which is divided into 11 intervalsof a fixed duration of 11 min each. The feed rates of thereagents are held constant within each interval (piece-wise constant input profiles). The surfactant feed rateand the VAc feed rate in the early intervals are varied,and the sensitivities with respect to each of thesevariables are analyzed. The surfactant feed rate in thefirst interval is constrained to lie between 0.83 and 4.16mL/min of 22.7 wt % aqueous solution. While the lowerlimit is fixed to enable considerable micellar nucleationat early times (there is no surfactant in the intialmixture), the upper limit is based on constraints thatallow secondary nucleation. A feed rate above this upperlimit would result in the nucleation of a very largenumber of particles initially, which upon growth wouldrender it impossible to cause the surfactant concentra-tion in the aqueous phase to exceed the critical micelleconcentration (which is a prerequisite for causing amicellar nucleation event), subject to the upper con-straints on the pumps. The feed rates in the remainingfour intervals are constrained only by the limits of thepumps. The VAc feed rate is likewise constrained to liebetween set limits. The feed rates of the BuA monomer

Figure 5. Effect of a perturbation in the initiator concentrationrelative to the base case experiment.

∂

∂tF(r,t) + ∂

∂r(F(r,t) drdt) )

Rcoag(r,t) + δ(r - rnuc)Rnuc(t) (1)


and the initiator components are fixed at nominalvalues. Different values of the feed rates of surfactantsolution and the VAc monomer in each of these intervalsare considered. Simulations are performed for all com-binations of the reagent feed rate values. Thus, a totalof 7776 simulations are performed in this combinatorialstudy. The end-point distributions produced in each ofthese cases are analyzed, by approximating as combina-tions of Gaussian distributions for the purposes ofcharacterization. Thus, each mode is characterized bya mean size, standard deviation in size, and the totalparticles in the mode. An analysis of the ratio of themean diameters of the bimodal distributions (among allthe distributions produced) shows a monotonic yetnonlinear dependence on the feed rate of the surfactantin each of these intervals. This suggests that thedistributions produced envelope all the attainable dis-tributions (subject to the discretization of the inputprofiles, the constraints imposed, and the modelingerrors). Thus, one can draw an envelope of the reachabledomains around these points, which characterize theexactly reachable distributions. Due to several reasonsincluding model uncertainties and measurement noise,it is advisable to allow a tolerance value on the attain-able proximity to the target distribution. Wang andDoyle III14 factor this aspect into their study by definingan ε-reachability, which is the reachability of distribu-tions within a defined tolerance ε (contrasted againstthe exact reachability). These ε-reachable domains canbe easily inferred from the exactly reachable distribu-tions identified in the present approach (by drawing acircle around each of the data points, the radius of whichrepresents the margin ε).

Figures 6 and 7 depict results pertaining to theseopen-loop simulations. Figure 6 characterizes the bi-modal distributions produced in these simulations interms of the means and the standard deviations of thetwo modes. This figure depicts the types of reachablebimodal distributions. It shows that as the size of thesmaller (secondary) mode increases, the standard devia-tion of this mode increases, resulting in a broad distri-bution (also with a reduced total particles in this mode).However, one can produce a larger average size in thelarger mode and still obtain a relatively lower standarddeviation. This case actually corresponds to clearlyseparated bimodal distributions (as seen in the right endof the plot in Figure 7a). Figure 7a shows a plot of thediameter ratio of the bimodal distribution, which showsrestrictions on the diameter of the primary mode for agiven diameter of the secondary mode. In general, asthe diameter of the smaller mode increases, the rangeof reachable diameters of the larger mode increases.

A suitable strategy for the control of distributions, aswas highlighted in the Experimental Section, is tocontrol the individual processes of nucleation, growth,and coagulation separately. Also, as was pointed outearlier, in emulsion polymerization, it is desirable tominimize the problematic coagulation events, utilizingthe nucleation and growth events as manipulatedvariables for the control of PSD. Thus, it is of interestto analyze the attainable distributions in the absenceof coagulation events. Figure 7b shows a plot of thediameter ratio of the attainable bimodal distributionsunder coagulation-free conditions, generated utilizinga coagulation-free model.19 In the absence of the coagu-lation events which contribute to discrete growth in theparticle size, the attainable size of the smaller mode is

much reduced. Also, the attainable range of size for thelarger mode is less sensitive to the size of the smallermode, and the size of the larger mode itself is also lower.

A direct parallel to the ideas of attainability regiondetermination described by other authors10,11,7 does notexist for this complex and intricate process. However,simulations were used to determine the lower and upperlimits in the attainable profiles of total particles andsolids content (see ref 25 for plots). These show that theprofile of solids content that can potentially be followedis relatively narrow, and the attainable end-point solidscontent lies in the range of 19-23.5% for the inputprofiles considered. These plots also show that the ratesdiminish toward the end of the batch, suggesting thatthese reachable distributions are retained beyond theend point of the distribution. The reachable distribu-tions are useful in setting proper targets for theoptimization problem involved in designing a recipe toachieve a target end-point distribution. Even though thetarget PSD is dictated by the end-use application of theemulsion latex, the reachability results aid in identify-ing if this target PSD is achievable and, if not, toidentify the achievable distribution that is closest to thetarget (the point that is closest to the target in thesefigures, based on a Euclidean distance, for example).Similarly, the profiles of the total particles and solidscontent aid in setting proper targets for the optimizationproblem involved in designing a recipe that tracks aparticular PSD trajectory in leading to a target end-

Figure 6. Bimodal distributions produced with surfactant andVAc monomer as the manipulated variables.


point PSD. Such a problem is presented in ref 21, inwhich the optimization problem was formulated in amultiobjective framework. A hierarchical control idea,based on the control of the individual nucleation,growth, and coagulation events, was employed. Also, thepartial control idea, based on the elimination or mini-mization of the coagulation events, was utilized. Thedistribution trajectory was re-cast as equivalent trajec-tories of nucleation and growth rates, which in turnwere converted into equivalent trajectories of totalparticles and solids content, employing the idea of thecontrol of instantaneous properties.

When the piecewise constant feed rates of surfactantalone is employed as design variables, the attainablelimits for the end-point solids content experiences anupward shift in the lower limit. However, the attainableregion of the distributions retains its shape (see ref 25for plots). This result suggests a potential for mid-coursecorrection, wherein an error introduced by the uncer-tainty in the feed rate of surfactant can be corrected bymanipulating the monomer feed rate and vice versa.This result is in perfect agreement with the findingsfrom the experimental studies presented previously, onthe interaction between nucleation and growth, and theeffects of surfactant and VAc on the process. To re-visitthe scenario that was presented among the experimen-tal results, a lower nucleation rate (caused by a lowerthan intended feed rate of the surfactant) would neces-sitate a decrease in the growth rate by reducing themonomer feed (to offset the interaction between nucle-ation and growth). Thus, one could prevent the particles

from growing to a larger size than the target, whichwould have occurred had no correction been made inthe monomer feed. But this correction in the size by adecrease in the monomer feed would result in a lowersolids content.

The sensitivity of these results to the input param-etrization (in particular, the duration of the zero-orderholds) was examined. It was seen that a larger intervalat early times results in very large primary nucleationrates, particularly at larger surfactant feed rates inthese intervals. This deprives the system of its abilityto cause a secondary nucleation event (for which it needsto breech the critical micelle concentration barrier). Thisclearly highlights the importance of allowing smallenough intervals in the process, particularly for open-loop optimization, in producing multimodal distribu-tions. One can also pose the problem of allowing theduration of the intervals as additional optimizationvariables.

It is pertinent to reiterate that, in this study, auniform gridding of the inputs was adopted, and everypossible combination of inputs were simulated, toidentify the reachable distributions. These distributionsidentified are exactly reachable, from which reachableregions based on different definitions can be deduced.However, one could instead use a statistical design ofexperiments approach26 to sample the reachable space(and identify similar exactly reachable distributions).This might lead to a considerable reduction in thenumber of simulations needed to be performed inarriving at the reachable distributions.

3.2. Robustness to Uncertainties and Distur-bances. The simulation studies presented above exam-ine the open-loop attainability of distributions in semi-batch processes. However, in addition to the inevitablemodel uncertainties, the process is characterized byseveral disturbances. This section examines the alter-ations to the attainable distributions introduced bythree most common disturbances: first, the latex car-ryover, second, the seeded polymerization (large latexcarry-over), and third, a model/parametric uncertaintywhich is rectified mid-course during the batch.

Batch-to-batch latex carryover and the associateduncertainty in the initial condition is a prevalent issuein industrial practice, motivating the analysis of itseffect on the reachable distributions. Figure 8 examinesthe effect of an uncertainty in the initial conditions onthe reachable distributions, based on an arbitrary initialdistribution (obtained from one of the experiments) withan initial solids content of less than 0.5%. Both VAc andsurfactant feeds were considered as manipulated vari-ables, and the coagulation effects were considered.Though most of the nominal reachable regions arecovered in this perturbed case, there are other possibledistributions that can be reached in the face of thisuncertainty in the initial condition (particularly withrespect to the secondary (smaller) mode, as seen inFigure 8c,d). The reason for the strip of large sizes inFigure 8b is that, at low surfactant feed rates at theearly intervals, the initial particles prevent any primarynucleation event by taking up most of the surfactants,and also grow rapidly with the high concentration ofthe monomers, until the nucleation of the secondarymode. The nucleation of the secondary mode is advancedin these cases, accounting for the larger particles (largerthan 150 nm) in the secondary mode (Figure 8c com-pared with Figure 8d). At higher feed rates of surfac-

Figure 7. Effect of coagulation on the diameter ratio of theattainable distributions with surfactant and VAc monomer as themanipulated variables.


tants in the initial intervals, the new particles nucleateddominate the particles in the initial batch, therebypreventing them from being evident in the end-pointdistribution. The implication of these results is thatrunning a batch in open loop with pre-optimized inputs,in the face of latex carryover, might result in consider-able difference in the resultant distribution comparedto the target. Thus, one needs to employ in-batchfeedback control, which can correct these errors anddrive the distribution toward the target. However, eventhough feedback can be used to bring the distributionback to the target (as seen by the nominal region lyingentirely within the perturbed region in Figure 8), thelarge particles are still present (in negligible quantity).In the perturbed case, the entire lower limit on theattainable profile of solids content is reduced relativeto the nominal case (Figure not shown). However, theupper limit essentially follows the nominal case exceptat the early times (where both limits are above thenominal case due to the initial particles). This again isa result that indicates that multiple manipulated vari-ables can bring the distributions back to the target (ina relative sense), although one might have to sacrificeperformance on the solids content tracking.

Figure 9 presents results to examine the effect of alarger mass of initial particles, but with a much reducedparticle sizes and a narrower distribution. This case canbe considered either as a larger initial disturbance withover 1% solids content or as seeded emulsion polymer-ization. Figure 9 shows the reachable bimodal distribu-tions in the perturbed case, while the reachable distri-butions corresponding to the nominal case are shown

in Figure 6. These results indicate an inverse effect fromwhat was seen in the previous case with a lower massof initial particles (latex carryover). In the current case,the perturbed distributions cover a smaller domain thanthe nominal case. Thus, in-batch feedback has a limitedutility in this case. For example, very large sizes of thelarger mode is not possible (Figure 9b compared withFigure 6b), as the seed particles add to the nucleatedparticles, thereby causing reduced growth rates andreduced particle sizes. The distributions with very largesizes in the larger mode (larger than 250 nm, Figure6b) seen in the nominal case correspond to low surfac-tant feed rates in the early intervals and large enoughfeed rates at the later intervals to cause a secondarynucleation event. The particles with the larger sizerange in the smaller mode (above 80 nm in Figure 6a)are also not possible due to the cascaded effect of theseed and growth on the nucleation event.

Figure 10 presents results to examine the effect ofan early disturbance in the process that is removed mid-course, on the reachable PSDs. The perturbation intro-duced could be due to several reasons, including modeland parameter uncertainties, implementation errors,and shear-induced coagulation, which could be a sto-chastic effect. This analysis is complementary to thelatex carryover-related initial condition disturbanceanalysis presented in the previous two cases, whichcannot be rectified mid-course (the particles carried overare always there). Figure 10a shows the perturbeddistribution relative to the nominal one, at 22 min inthe batch. This corresponds to a time when a nucleationevent is underway (as seen by the large peak of particles

Figure 8. Effect of the initial distribution (of small mass) on the reachable bimodal distributions.


in the smallest end). Figure 10b shows the end-pointdistributions that are reached in the nominal case.Assuming that after 22 min there are no furtherstochastic effects, or that the source of the uncertaintythat caused the deviation in the distribution at 22 minhas been removed, the resultant distributions at the endof the batch are shown in Figure 10c, which shows avery profound difference when compared to Figure 10b,even for such a small variation. (Note that all the otherprocess states are assumed to be unaltered between thenominal and perturbed cases). Figure 10d characterizesthe attainable bimodal distributions in terms of themean and the diameter ratio. The nominal reachableregion is much more concentrated than the reachableregion corresponding to the perturbed distribution.Given the monotonicity with respect to the inputs’, onecould conclude that at least part of the nominal regionlies within the perturbed region. This region of intersec-tion is amenable to feedback correction, while a targetin the complementary region (to the perturbed region)cannot be attained after this mid-course disturbance.In the latter case, if the source of this uncertainty is adeterministic disturbance, then the best recourse is toembark on a batch-to-batch control strategy.27,28 On theother hand, if it is caused by stochastic effects, one hasto perform robust design, say based on worst-caseconsiderations.29

4. SummaryThe sensitivities in the process of the evolution of PSD

in semibatch emulsion copolymerization using nonionic

surfactants were studied via experiments and simula-tions. The experimental studies indicate the following:

(1) The need for multiple process inputs for the controlof PSD.

(2) Surfactant and a monomer (in this case theprimary monomer VAc) as the inputs, particularly whenemploying nonionic surfactants.

(3) The suitability of a hierarchical control strategyin which the individual rates of nucleation, growth, andcoagulation are controlled to produce the desired com-plete distribution.

(4) Irreversibility in several aspects of the processswhile some of these such as the skewness of thedistributions may not be critical depending upon theend-use applications, others such as the effect of a largergrowth rate or a stronger coagulation rate might leavean intolerable deviation in the distribution.

(5) Inherent limitations within the process thatrestrict the type of distributions that can be produced.

(6) The limitations in the process that in some casestranslate into lower solids content (and hence have abearing on the economy of the process), in employingfeedback to correct the correctable errors in the distri-bution.

(6) The need for a detailed first-principle model to beemployed for the open-loop and closed-loop control ofPSD.

The simulation studies were used to gain furtherinsights into the potentials and limitations in theprocess and to determine the type of distributions thatcan actually be produced considering system and ex-ternal limitations. A controllability analysis based on alinearized model is unsuitable due to the discontinuityin the process and its highly nonlinear character. Arigorous mathematical analysis of the reachability andcontrollability is also beyond the reach of this process,due to the underlying complexity. Thus, a simplesimulation-based analysis was performed, bearing inmind the practicality or the operability of the process.The study gave insight into the restrictions on the typesof distributions that can be produced and also revealedthe effect on these restrictions under different controlconfigurations. The effect of a partial control configu-ration (with coagulation not being recruited as a ma-nipulated variable for control of PSD) was examined.The total control configuration, which controls all thethree major subprocesses, was seen to produce a muchwider class of distributions, even though this configu-ration is subject to questions about the feasibility ofenabling tight control on the coagulation events. Thestudy identified the types of exactly reachable distribu-tions (currently restricted to a low solids regime). Keyaspects of the distributions (such as the mean diameterratio of bimodal distributions) were found to have amonotonic (though nonlinear) dependence on the inputs.On the basis of this observation, reachability domainsare determined by envelopes around these reachablepoints. A particular discretization of the inputs alongthe batch and a fixed reaction time were considered inthis study. Relaxation of these restrictions were exam-ined, which gave further insight into the formulationof the problem of optimization and control of PSD.

Further, the effect of uncertainties and disturbanceson the reachable distributions were analyzed. Theresults reveal two different scenarios:

(1) One in which in-batch feedback control is feasibledespite the internal limitations (irreversibility) and

Figure 9. Comparison of the reachable bimodal distributions inthe seeded case with those in the nominal (ab initio) case (Figure6).


external limitations (constraints, sparse and delayedmeasurements, etc.).

(2) Another in which in-batch feedback control mightnot be effective in all cases in correcting the errors,thereby advocating batch-to-batch control.

These situations suggest a combination of in-batchand batch-to-batch feedback control strategy for thecontrol of PSD in semibatch emulsion polymerization.There is also enormous potential for robust optimizationformulations, which take explicit account of potentialuncertainties and disturbances in performing a conser-vative open-loop optimal design of the process.

Acknowledgment

The authors acknowledge collaboration and supportfrom Dr. Cajetan F. Cordeiro of Air Products andChemicals Inc. and Dr. Yang Wang at the Universityof California, Santa Barbara. Financial support from theUniversity of Delaware Competitive Fellowship, theOffice of Naval Research, and the University of Dela-ware Process Control and Monitoring Consortium arealso gratefully acknowledged.

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Received for review February 18, 2003Revised manuscript received June 2, 2003

Accepted June 6, 2003

IE030145P


a sensitivity approach to reachability analysis for particle size distribution in semibatch emulsion...

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