AMPS/AAm/AAc Terpolymerization: Experimental Verification of the EVM Framework for Ternary Reactivity Ratio Estimation

Transcription

1 processes Article AMPS/AAm/AAc Terpolymerization: Experimental Verification EVM Framework Ternary Reactivity Ratio Estimation Alon J. Scott, Niousha Kazemi Alexer Penlid * Institute Polymer Research (IPR), Department Chemical Engeerg, University Waterloo, Waterloo, Ontario, N2L G1, Canada; ajscott@uwaterloo.ca (A.J.S.); nkazemi@uwaterloo.ca (N.K.) * Correspondence: penlid@uwaterloo.ca; Tel.: (ext. 664) Academic Edir: Michael Henson Received: 5 January 2017; Accepted: 21 February 2017; Publhed: 25 February 2017 Abstract: The complete error--variables-model (EVM) framework, constg both design experiments parameter estimation stages, applied ization 2-acrylamido-2-methylpropane sulfonic acid (AMPS, M 1 ), acrylamide (AAm, M 2 ) acrylic acid (AAc, M ). Th water-soluble has potential applications enhanced oil recovery, but associated ization ketic charactertics largely unstudied. In current paper, EVM used design experiments ( first time literature), s subsequently estimated based on both low medium-high conversion data. The results from medium-high conversion data more prece than those from low conversion data, ree used next predict trajecry over full course conversion. Good agreement seen between data model predictions, which confirms accuracy newly determed ternary s: r 12 = 0.66, r 21 = 0.82, r 1 = 0.82, r 1 = 0.61, r 2 = 1.61, r 2 = Keywords: 2-acrylamido-2-methylpropane sulfonic acid; acrylamide; acrylic acid; error--variables-model; polymerization ketics; estimation; ization 1. Introduction Water-soluble s have applications a wide variety as such as enhanced oil recovery (EOR), dewaterg, meral processg flocculation. Most se applications rely on fact that addition polymeric material can alter rheology an aqueous medium [1]. Among syntic water-soluble polymers, polyacrylamide used as a base many applications. It ten difficult one sgle polymer meet all requirements an application, so copolymers s can be employed deliver specific properties. One most widely used copolymers acrylamide acrylamide/acrylic acid (AAm/AAc) copolymer. The AAm/AAc copolymer can be used many above-mentioned applications, cludg enhanced oil recovery [2]. However, it has been observed that AAm/AAc copolymer degrades hostile environments (typical EOR conditions). Theree, addition a third comonomer resultg a backbone with higher rmal shear stability has been suggested overcome th problem. One such comonomer that can enhance stability AAm/AAc copolymer harsh environments 2-acrylamido-2-methylpropane sulfonic acid (AMPS). AMPS a larger monomer molecule compd AAm AAc, which, when corporated, provides better rmal shear stability, as a result, makes fal polymer more suitable EOR applications []. Processes 2017, 5, 9; doi:10.90/pr

2 Processes 2017, 5, The AMPS/AAm/AAc a largely unstudied system that has only apped literature with past ten years. Some applications th new have been reported EOR [], oil-field drillg [4], superabsorbent hydrogels [5], sludge dewaterg [6], controlled drug-delivery systems [7]. In se few studies, only fal properties substrate (such as swellg behavior, restance temperature shear stress) have been dcussed, but ketic charactertics ization have not been reported. Sce fal application properties th directly related its microstructure, it essential have a clear understg ization ketics. Given that re different possibilities termal monomer (on growg radical), options added monomer, ne different propagation steps possible accordg termal model: M 1 + M k 11 1 M1 M 1, M 1 + M 2 M 1 + M M 2 + M 1 M 2 + M 2 M 2 + M M + M 1 M + M 2 M + M k 12 M1 M 2, k 1 M1 M, k 21 M2 M 1, k 22 M2 M 2, k 2 M2 M, k 1 M M 1, k 2 M M 2, k M M. In th series reactions, M i represents a radical species with monomer i at cha end (i = 1, 2, ). Similarly, M j represents monomer j that beg added cha end (j = 1, 2, ). Each ne reactions has a rate constant, k ij (radical i addg monomer j). Six parameters, called monomer s (r ij ), can be used describe potential homopropagation relative potential cross-propagation. r 12 = k 11 k 12, r 1 = k 11 k 1, r 2 = k 22 k 2, r 21 = k 22 k 21, r 1 = k k 1, r 2 = k k 2. Reactivity s crucial study ketics multicomponent polymerization systems. Terpolymerization systems frequently utilized dustry possess valuable mation academic research, yet re a considerable lack estimation studies such systems. Th partially due complexity model, Alfrey Goldfger model (Equation (1)). F i stantaneous mole fraction monomer i corporated (bound), r ij monomer s relatg i j, f i correspondg mole fraction unreacted (free) monomer i (ten referred as feed mole fraction). Equation (1) relates stantaneous (not cumulative) copolymer properties: F 1 F 2 = F 1 F = F 2 F = ( f 2 ( f ( f2 f )( )( /r 1 r 21 + f 2 /r 21 r 2 + f /r 1 r 2 + f 2 /r 12 + f /r 1 f 1 /r 12 r 1 + f 2 /r 12 r 2 + f /r 2 r 1 f 2 + f 1 /r 21 + f /r 2 )( )( /r 1 r 21 + f 2 /r 21 r 2 + f /r 1 r 2 + f 2 /r 12 + f /r 1 f 1 /r 1 r 21 + f 2 /r 2 r 12 + f /r 1 r 2 f + f 1 /r 1 + f 2 /r 2 )( )( /r 12 r 1 + f 2 /r 12 r 2 + f /r 2 r 1 f2 + f 1 /r 21 + f /r 2 f 1 /r 1 r 21 + f 2 /r 2 r 12 + f /r 1 r 2 f + f 1 /r 1 + f 2 /r 2 ), (1a) ), (1b) ). (1c)

3 Processes 2017, 5, 9 16 However, more importantly, knowledge gap ization ketics related a de fac accepted analogy between copolymerization ization mechanms; researchers ten use s obtaed bary pairs (from copolymerization experiments) ization models. However, because error bary data tends propagate ternary system, bary s cannot be used describe ternary systems; at best, th provides an approximation [8]. More importantly, ternary s never determed usg ternary data, differences system make it imprudent use bary ternary s terchangeably. Usg appropriate s may affect model permance predictg ( sequence length charactertics, sce se also depend on values) determation or ization charactertics (such as azeotropic pot). Despite se rks, all studies permed previously have employed bary s directly models. Th should be avoided [8]. Problems associated with estimation design experiments systems have largely been resolved usg error--variables-model (EVM), which was dcussed recently by Kazemi et al. [9] ( will be reviewed briefly current paper). Thus, what follows, AMPS/AAm/AAc vestigated by implementg EVM framework accurate determation ternary s. Parameter estimation implementation design experiments strategy demonstrated, estimates analyzed terms both precion accuracy. Fally, values based on ly selected experiments suggested. Comparons between low medium-high conversion level data also cluded exame effect data set ( its herent errors) on estimation results. 2. Experimental 2.1. Design Experiments Optimal design experiments leads creased mation content while mimizg number experiments obtag more prece parameter estimates. Under error--variables-model (EVM) framework, one can design experiments that consider error all variables volved (both dependent dependent) process model [9,10]. A brief guide estimatg reliable ternary s from ization data provided Figure 1, a comprehensive evaluation with detailed explanations has been previously publhed by Kazemi et al. [8,9]. Additional comments related steps Figure 1 provided below: 1. Review literature any mation on polymerization ketics system question. 2. Use literature values (if any) prior knowledge fd (or guess) reasonable prelimary s system; determe wher re any constrats on feed s. 2.a. In absence any prior mation, run prelimary experiments. Each should have a rich one monomers (e.g., 80% or higher), any limitations (constrats) feasible region should be cfully recorded. Estimate prelimary ternary s.. Choose feed s accordg EVM framework. Each recipe should conta 80% one monomer an equal amount or two (that, f i /f j /f k : (0.8/0.1/0.1), (0.1/0.8/0.1) (0.1/0.1/0.8)). If a polymerization recipe contag 80% one monomer not achievable (due possible feed constrats), choose lower percentages as necessary. 4. Perm experiments at low conversion ( 5% 10%) /or up medium-high conversion levels (50% 70%). Collect data correspondg conversion values. 5. Use EVM parameter estimation methodology estimatg s construct jot confidence regions (JCRs). Refer work by Kazemi et al. [8,10] a detailed implementation th method.

4 Processes 2017, 5, If satfied with precion results, move next step. If not satfied, perm dependent replicates experiments re-estimate s. 7. Present estimates ir jot confidence regions. Processes 2017, 5, Th Th methodology methodology used used ization ization AMPS/AAm/AAc; AMPS/AAm/AAc; details details each each step step presented presented what what follows. follows. For For first first time time literature, literature, EVM framework EVM framework ternary ternary estimation estimation verified ly. verified ly. Figure 1. Flowchart EVM (error--variables-model) framework ternary Figure 1. Flowchart EVM (error--variables-model) framework ternary estimation. estimation Reagent Purification Monomers 2-acrylamido-2-methylpropane sulfonic acid (AMPS; 99%), acrylamide (AAm; electrophores grade, 99%), acrylic acid (AAc; 99%) were purchased from Sigma-Aldrich

5 Processes 2017, 5, Reagent Purification Monomers 2-acrylamido-2-methylpropane sulfonic acid (AMPS; 99%), acrylamide (AAm; electrophores grade, 99%), acrylic acid (AAc; 99%) were purchased from Sigma-Aldrich (Oakville, Ontario, Canada). AAc was purified via vacuum dtillation at 0 C, while AAm AMPS were used as received. Initiar (4,4'-azo-b-(4-cyanovaleric acid), ACVA), hibir (hydroquone) sodium hydroxide were also purchased from Sigma-Aldrich. Sodium chloride from EMD Millipore (Ebicoke, Ontario, Canada) was used as received. Water was Millipore quality (18 MΩ cm), acene methanol were used as received from suppliers. Nitrogen gas (4.8 grade) purchased from Praxair (Mssauga, Ontario, Canada) was used degassg solutions. 2.. Polymer Syns Aqueous monomer solutions with a monomer concentn 1 M an itiar (ACVA) concentn M (relative tal solution volume) were prepd. Specific feed s (that, pre-polymerization solution s) will be dcussed Step Results Dcussion section (see Section.). As demonstrated a recent study [11], constant ph ionic strength extremely important water-soluble copolymer syns. Theree, solutions were titrated with sodium hydroxide adjust ph approximately 7 (±0.5). Similarly, ensure constant ionic strength (IS) between experiments, sodium chloride was added each pre-polymerization solution. The highest IS occurs when solution rich AMPS, so sodium chloride was added all or pre-polymerization solutions reach that same IS value. The solutions were n purged with 200 ml/m nitrogen 2 h. After degassg, aliquots ~20 ml solution were transferred sealed vials usg cannula transfer method [12]. Terpolymerizations were run a temperature controlled shaker-bath (OLS200; Grant Instruments, Cambridge, UK) at 40 C 100 rpm. Vials were removed at selected time tervals, placed ice furr jected with approximately 1 ml 0.2 M hydroquone solution sp polymerization. Polymer samples were olated by precipitatg products acene or methanol, filtered (paper filter grade number 41, Whatman; Sigma-Aldrich, Oakville, Ontario, Canada) vacuum dried 1 week at 50 C. All polymerizations were dependently replicated Polymer Characterization Conversion polymer samples was determed usg gravimetry. The mass sodium ions was also considered conversion calculations, as per recommendation Riahezhad et al. [11,1]. Copolymer was measured usg elemental analys (CHNS, Vario MICRO Cube, Elementar, Isospark Canada Inc., Montreal, Canada). Calculation did not clude H content, as residual water has been known affect determed H content [12]. Select samples were dependently replicated.. Results Dcussion The error--variables-model framework, which was outled Figure 1, applied ization AMPS/AAm/AAc. Th first time literature that entire framework, from prelimary vestigation design experiments parameter estimation, verified ly..1. Step 1: Review Literature Polymerization Ketics In recent years, several studies have vestigated AMPS/AAm/AAc. These have focused on syns, characterization, potential applications th ; none studies have cluded ization ketics.

6 Processes 2017, 5, For example, Bao et al. [5] grafted AMPS/AAm/AAc on sodium carboxymethyl cellulose montmorillonite (MMT) create a superabsorbent hydrogel. In th case, physical properties synsized (such as degree swellg, water retention, morphology) were focus analys. Similarly, Ma et al. [6] synsized AMPS/AAm/AAc via UV irradiation use as a flocculent. While th group did provide more mation about ir syns steps, overall focus paper was applications. Polymer charactertics cludg trsic vcosity, dsolution time flocculation permance were presented. Th particular has also been used drug-delivery applications [7]. The drug-delivery system uses superabsorbent polymer composites, so charactertics such as swellg capacity drug encapsulation efficiency were studied. While vestigation cluded release priles drug-delivery, it did not dcuss details surroundg polymerization ketics. In perhaps most relevant papers current work, Peng et al. [4] Zaiun et al. [] have studied AMPS/AAm/AAc petrochemical applications. The work by Peng et al. [4] describes free-radical ization AMPS/AAm/AAc its application as a high-temperature restant filtn control agent. Zaiun et al. [] have vestigated potential use AMPS/AAm/AAc enhanced oil recovery (EOR) applications, as AMPS comonomer expected improve shear stability limit rmal degradation (compd stard AAm/AAc copolymers). However, both se cases, authors make no mention polymerization ketics. The ketic charactertics beg synsized directly related its microstructure. Theree, it important have a clear understg ization ketics. Sce th mation not available literature, reliable s th AMPS/AAm/AAc system will be determed ly what follows..2. Step 2: Determe Prelimary Reactivity Ratios Establh Feed Composition Constrats Sce, date, re have been no ketic studies th particular ization literature, bary values associated copolymer pairs were used as prelimary s. These bary values were obtaed ly, which allowed same set-up be used both co- izations. For AMPS/AAm AMPS/AAc copolymerizations, an -depth study was recently completed by Scott et al. [14]. Different estimates se copolymers were publhed previously, but most se estimates were subject numerous sources error (namely, lear parameter estimation techniques non-lear parameter estimation with no dependent replication). In an attempt provide most accurate estimates possible, Scott et al. [14] used error--variables-model (EVM) technique design experiments estimate s both AMPS/AAm AMPS/AAc copolymerizations. As an additional advantage, polymerization conditions (ph, ionic strength, etc.) that were used se copolymerizations also used izations described current paper. Similarly, AAm/AAc copolymerization, Riahezhad et al. [12] conducted a thorough vestigation s th copolymerization system. Aga, same polymerization conditions were used AAm/AAc bary system izations described current paper. Theree, s arrived at above bary systems can confidently be considered as best bary s different pairs. These values summarized Table 1. Table 1. Bary s copolymerizations associated with 2-acrylamido-2-methylpropane sulfonic acid (AMPS) 1 /acrylamide (AAm) 2 /acrylic acid (AAc). Source T( C) ph r 12 r 21 r 1 r 1 r 2 r 2 Scott et al. [14] Riahezhad et al. [12]

7 Processes 2017, 5, One advantages associated with EVM ability troduce feed constrats on design. However, sce very little ketic mation available AMPS/AAm/AAc ization, it difficult establh wher such constrats ext. In studyg AMPS/AAc copolymer, Scott et al. [14] reported that polymerization was extremely slow mimal precipitate med when AAc fraction was high feed (f AAc,0 = 0.85). Similarly, Ryles Neff [1] observed that a prelimary feed f AAc,0 = 0.80 AMPS/AAc copolymer made polymer olation ( subsequent filterg/precipitation) difficult as a result phase sepan. Theree, AMPS/AAm/AAc, feed AAc was constraed such that f AAc, Step : Apply Design Experiments Select Optimal Feed Compositions The basic idea behd design experiments select feed s ( trials) which mimize variability parameter estimates. As mentioned previously, ization studies ten (correctly!) use s extracted from bary systems; based on th analogy, a ternary system treated as separate bary copolymerizations. Th approach approximate at best, as it propagates herent error present bary estimates, overlooks effect teractions between all monomers on ir wards each or not at all reliable predictg ternary s. An additional problem with ization studies m Alfrey Goldfger (A G) model that typically used evaluate stantaneous ization (see Equation (1)). Kazemi et al. [15] recently showed that selectg different combations s mole fractions A-G model (e.g., F 1 /F 2 F 1 /F versus F 1 /F 2 F 2 /F ) can affect precion estimates. Th work exposed fact that model suffers from symmetry sues; fal results depend on arbitrary choice different combations copolymer mole fractions parameter estimation scheme. Theree, current future ization vestigations, a recast version model (courtesy Kazemi et al. [8]) should be used. The recast A G model presents each stantaneous mole fraction as a sgle response (see Equation (2)). While se expressions may seem more complex than conventional A G model, th mulation symmetrical error structures not drted [8]: F 1 = ( f 1 r 21 r + f 2 1 r 21 r + f )( 2 r 1 r f 1 + f 2 2 r + f ) 12 r ( 1 f 1 r 21 r + f 2 1 r 21 r + f )( 2 r 1 r f 1 + f 2 2 r + f ) ( 12 r + f 2 1 r 21 r + f 2 1 r 12 r + f )( 2 r 1 r f 2 + f 1 2 r + f ) ( 21 r + f 2 r 1 r + f 2 21 r 2 r + f )( 12 r 1 r f + f 1 2 r + f ), 2 1 r 2 (2a) F 2 = F = ( f 2 r 21 r + f 2 1 r 12 r + f )( 2 r 1 r f 2 + f 1 2 r + f ) 21 r ( 2 f 1 r 21 r + f 2 1 r 21 r + f )( 2 r 1 r f 1 + f 2 2 r + f ) ( 12 r + f 2 1 r 12 r + f 2 1 r 12 r + f )( 2 r 1 r f 2 + f 1 2 r + f ) ( 21 r + f 2 r 1 r + f 2 21 r 2 r + f )( 12 r 1 r f + f 1 2 r + f ), 2 1 r 2 ( f r 1 r + f 2 21 r 2 r + f )( 12 r 1 r f + f 1 2 r + f ) 1 r ( 2 f 1 r 21 r + f 2 1 r 21 r + f )( 2 r 1 r f 1 + f 2 2 r + f ) ( 12 r + f 2 1 r 12 r + f 2 1 r 12 r + f )( 2 r 1 r f 2 + f 1 2 r + f ) ( 21 r + f 2 r 1 r + f 2 21 r 2 r + f )( 12 r 1 r f + f 1 2 r + f ). 2 1 r 2 (2b) (2c) With th new mation md, goal th step procedure apply EVM design criterion recast Alfrey Goldfger model so that feed s (that can lead most reliable s) selected. EVM considers error all terms, so usg a design experiments technique with EVM context helps account error both dependent variables (feed s) dependent variables ( s). Details have been presented previously [8,9], but key pots briefly revited below. The EVM design criterion aims maximize determant mation matrix (G), which verse variance covariance matrix parameters: G = n r i Z i( Bi VB 1Zi i), () i=1

8 Processes 2017, 5, where r i = number replicates at i th trial (out n trials), Z i = vecr partial derivatives model function with respect parameters ( th case, partial derivatives recast A G model (Equation (2)) with respect s), B i = vecr partial derivatives model function with respect variables (aga, th case, partial derivatives recast A-G model (Equation (2)) with respect feed (f ) (F) s), Processes V = 2017, variance covariance 5, 8 matrix variables (which provides mation about measurement 8 16 error possible correlation variables volved). As As explaed explaed previous previous work work by by Kazemi Kazemi et et al. al. [9], [9], experiments experiments sufficient sufficient estimate estimate ization ization s s th th nonlear nonlear model model scenario. scenario. In In ization ization problem, problem, EVM EVM model model consts consts equations equations (see (see aga aga Equation Equation (2)) (2)) five variables five variables (, f2, F1, (f 1, F2, f 2, F); F 1, only F 2, F two ); only two feed s feed s dependent (f dependent = 1 (f f2) = 1 f 1 f 2 ) s s measured measured dependently. dependently. Theree, Theree, ization, ization, re re two dependent two dependent variables variables (5 variables (5 variables equations equations = 2) = 2) six (6) six parameters (6) parameters ( ( s). s). The number The number experiments experiments needed needed can be can calculated be calculated by dividg by dividg number number parameters parameters by by number number dependent dependent variables variables (see (see Bard Bard [16] [16] Duever Duever et al. et [17]); al. [17]); hence, hence, =. 2 =. For For ternary ternary estimation, estimation, feed feed s s typically typically located located at at corners corners triangular triangular (ization) (ization) plot plot [9]. [9]. To To help help vualize vualize location location feed feed s s th th system, system, Figure Figure 2 combes combes sub sub regions regions (shaded (shaded as) as) as as well well as as pots pots located located close close or or side side se se regions. regions. Polymerizations Polymerizations run run usg usg se se feed feed s s (recipes) (recipes) will will provide provide sufficient sufficient mation mation reliable reliable estimation. estimation. Figure2. 2. Optimal feed regions s 2-acrylamido-2-methylpropane sulfonic acid (AMPS)/acrylamide (AAm)/acrylic acid acid (AAc) (AAc) ization Step Step4: 4: Perm Perm Experiments; Collect Collect Conversion CompositionData.4.1. Low Conversion Experiments The first attempt at estimatg s AMPS/AAm/AAc ization was conducted by by analyzg low low conversion conversion data, data, similar similar conventional conventional approaches approaches estimatg estimatg s copolymerizations s copolymerizations izations izations [9,18]. As[9,18]. shownas shown Table 2, Table 2, feed s feed s correspond correspond feed s feed s Figure 2 Figure (reflectg 2 (reflectg also process also process constrats), constrats), data were limited data were lowlimited conversion low (X w < conversion 0.100). Conversion (Xw < 0.100). values Conversion values with a * dicate results from an dependently replicated polymerization; same classification will be used Table. That, se replicates were synsized entirely dependently, usg freshly made solutions, etc. Conversion was determed aga usg gravimetry, samples were dependently characterized usg elemental analys, as described Section 2.4. Note also that data Tables 2 presented terms mass conversion (Xw), which should not be confused with molar conversion (Xn; see, example,

9 Processes 2017, 5, with a * dicate results from an dependently replicated polymerization; same classification will be used Table. That, se replicates were synsized entirely dependently, usg freshly made solutions, etc. Conversion was determed aga usg gravimetry, samples were dependently characterized usg elemental analys, as described Section 2.4. Note also that data Tables 2 presented terms mass conversion (X w ), which should not be confused with molar conversion (X n ; see, example, Equations (4) through (6)). Table 2. Experimental data AMPS/AAm/AAc ization; low conversion. Optimal Feed Composition #1 #2 # X w f AMPS,0 f AAm,0 f AAc,0 F AMPS F AAm F AAc * * * * * * * * * In th Table, X w = conversion, f i,0 = itial feed (monomer i) F i = cumulative. Sce conversion level was kept low se runs, it can be assumed that drift negligible. Theree, stantaneous model (that, recast Alfrey Goldfger model, Equation (2)) EVM parameter estimation technique were employed [9]. Details regardg parameter estimation technique, resultg estimates, correspondg jot confidence regions (JCRs) will be presented Step 5 (Section.5) Medium-High Conversion Experiments The recast A G model (Equation (2)) a significant improvement over Equation (1), but it only valid low conversion data sets. In order copolymer drift be negligible (that, itial feed rema unchanged (measurable) cumulative copolymer be equal its stantaneous value), data must be collected at very low conversion levels. Th restrictive assumption troduces additional sources error, cludg significant difficulties. As an alternative, a cumulative ternary model has been considered order estimate ternary s usg full conversion trajecry. The cumulative model (essentially Sket equation applied ization), shown Equation (4), relates cumulative each monomer (F i ) itial mole fraction monomer feed (f i,0 ), correspondg mole fraction unreacted monomer (f i ) molar conversion (X n ): F 1 = f 1,0 f 1 (1 X n ) X n, (4a)

10 Processes 2017, 5, F 2 = f 2,0 f 2 (1 X n ) X n, (4b) F = f,0 f (1 X n ) X n. (4c) In th step procedure, s AMPS/AAm/AAc ization were estimated usg same feed s Figure 2, but th time runng izations medium-high conversion levels (see Table ). Sce it no longer valid assume constant (that, drift no longer negligible), f i must be evaluated over conversion X n, accordg model ordary differential equation m, shown Equation (5) (where F i values calculated usg Equation (2)). Given itial conditions f i = f i,0 at X n = 0, a numerical solution can be used evaluate s along full conversion trajecry: d f 1 dx n = f 1 F 1 1 X n, (5a) d f 2 dx n = f 2 F 2 1 X n, (5b) d f dx n = f F 1 X n. (5c) It important note that molar conversion (X n ) used both Equations (4) (5), but that mass conversion (X w ) reported data tables (see Tables 2 ). Molar conversion mass conversion related usg monomer molecular weights (MW i ), as shown Equation (6): X n = X w MW 1 f 1,0 + MW 2 f 2,0 + MW f,0 MW 1 F 1 + MW 2 F 2 + MW F. (6) Th methodology (usg direct numerical tegn (DNI) evaluate cumulative model) has been described previously by Kazemi et al. [8]. The current approach (i.e., tegratg stantaneous model (Equation (2)) over conversion via Equations (4) (5) conductg parameter estimation via EVM simultaneously) generally preferable parameter estimation, as it cludes all available mation from system (not only at low conversion as per typical approaches), does not suffer from limitg assumptions or difficulties associated with low conversion data analys. As was case low conversion experiments, estimation details along with estimates correspondg JCRs will be shown Step 5 (Section.5). Table. Experimental data AMPS/AAm/AAc ization; medium-high conversion. Optimal Feed Composition #1 X w f AMPS,0 f AAm,0 f AAc,0 F AMPS F AAm F AAc * * * * *

11 Processes 2017, 5, Table. Cont. Optimal Feed Composition #2 # X w f AMPS,0 f AAm,0 f AAc,0 F AMPS F AAm F AAc * * * * * * * * * * * * * * * * Step 5: Use EVM Estimate Reactivity Ratios; Construct Jot Confidence Regions The error--variables-model (EVM), described previously design experiments (see Section.), can also be used current parameter estimation step. EVM one most powerful non-lear regression approaches available, as it considers all sources error (both dependent dependent variables) [19]. EVM not only ces experimenter consider all sources error, but also provides estimates true values or variables volved model along with parameter estimates. Theree, it by far most stattically correct comprehensive approach estimation [20]. In th step, EVM approach used estimate s both low medium-high conversion data. However, as dcussed Section.4, ization model differs each data set: low conversion data analyzed usg recast stantaneous ization model (Equation (2)) along with related low conversion assumptions, whereas medium-high conversion data analyzed with direct numerical tegn (DNI) stantaneous model, i.e., usg cumulative model (see Equations (2), (4) (5)). The nested-iterative EVM implementation has been described detail several previous references ( stance, see references [8 10,17,18,21,22] cited here), so no furr details will be presented. The ization estimates correspondg JCRs both data sets (Tables 2 ), along with correspondg bary (copolymerization) s (Table 1) presented Figure. In all cases, results show that JCRs from medium-high conversion data smaller ( ree more prece) than JCRs from low conversion data. These results as

12 Processes 2017, 5, expected; utilizg all mation available elimatg potentially accurate assumptions can improve precion pot estimates. These results confirm ( first time) that Processes EVM2017, can 5, 8 successfully be used analyze directly data from izations 12 over 16 whole conversion range. In addition, results prove that ly designed ternary feed s ly designed can provide ternary sufficient feed mation s estimate can provide sufficient s with mation very little correlation estimate th ization s with very system, little correlation as one can realize th ization from orientation system, as JCRs. one can In realize addition, from all cases, orientation literature JCRs. bary In addition, all estimates cases, literature locatedbary outside ization estimates JCRs. located outside ization JCRs Bary (Prelimary) (0.18, 0.85) Low Conversion Ternary (0.69, 0.81) Medium-High Conversion Ternary (0.66, 0.82) r AAm/AMPS r AMPS/AAm (a) Bary (Prelimary) (0.19, 0.86) Low Conversion Ternary (0.91, 0.68) Medium-High Conversion Ternary (0.82, 0.61) r AAc/AMPS r AMPS/AAc (b) Figure. Cont. Figure. Cont.

13 Processes 2017, 5, Processes 2017, 5, Bary (Prelimary) (1., 0.2) Low Conversion Ternary (1.76, 0.27) Medium-High Conversion Ternary (1.61, 0.25) 0. r AAc/AAm r AAm/AAc (c) Figure.. Reactivity Reactivity estimation estimation results results AMPS/AAm/AAc AMPS/AAm/AAc ization ization (with prelimary (with copolymerization prelimary copolymerization estimates fromestimates literature from [12,14]). literature [12,14]) Step Step 6: 6: Decide Decide Wher Wher Results Results Are Are Prece Prece Enough Enough In In Step Step 5 (Section (Section.5),.5), estimates estimates associated associated JCRs JCRs confirm confirm that that EVM-based EVM-based design design parameter parameter estimation estimation method method ternary ternary systems systems work work very very well well with with data data directly directly from from AMPS/AAm/AAc AMPS/AAm/AAc ization. ization. The The reliability reliability results results first first establhed establhed by by examg examg size size JCRs JCRs by by notg notg lack lack correlation correlation between between parameters parameters (see (see Figure Figure ). ). In In second second diagnostic diagnostic stage, stage, it it important important vestigate vestigate accuracy accuracy s s by runng by runng additional additional checks. checks. One One most most common common diagnostic diagnostic checks checks evaluate evaluate behavior/priles behavior/priles cumulative cumulative.. Model Model predictions predictions (usg (usg estimates) estimates) over over polymerization polymerization trajecry trajecry compd compd ly ly measured measured s. s. An An acceptable acceptable agreement agreement between between predicted predicted results results reflects reflects reliability reliability accuracy accuracy s s ization ization system. system. Thus, Thus, direct direct numerical numerical tegn tegn (DNI) (DNI) was was applied applied recast recast version version stantaneous stantaneous equation equation (see (see Equation Equation (2)) usg (2)) newly usg determed newly determed estimates. Sce estimates. medium-high Sce medium-high conversion data conversion provided data more provided mation more ( mation smaller JCRs), ( smaller JCRs), s estimated s from estimated data Table from were data used. The Table predicted were cumulative used. The predicted cumulative trajecries versus conversion, trajecries as well versus as conversion, as well pots as obtaed via elemental pots analys, obtaed via shown elemental Figure analys, 4 all shown Figure ly 4 all designed feed ly s. designed feed s. Figure Figure 4 shows shows that, that, all all cases, cases, predicted predicted trajecries trajecries (from (from ternary ternary estimates) estimates) capture capture ly ly observed observed behavior behavior satfacrily. satfacrily. At At low low conversion, conversion, however, however, re re some some mor mor dcrepancies dcrepancies between between model model results. results. The The noe noe seen seen data data typical typical at at such such low low conversions, conversions, which which confirms confirms need need higher higher conversion conversion experiments. experiments. Thus, Thus, spite spite natural natural variation variation results, results, it it possible possible conclude conclude that that cumulative cumulative (DNI) (DNI) model model EVM-based EVM-based estimated estimated ternary ternary s s can can successfully successfully predict predict behavior behavior system. system. Th an important diagnostic check th system ( ization studies, general), Th an important diagnostic check th system ( ization studies, general), as it dicates that employg ization model EVM framework leads prece reliable s system.

14 Processes 2017, 5, as it dicates that employg ization model EVM framework leads prece Processes 2017, reliable 5, 8 s system Cumulative Composition AMPS (Experimental) AAm (Experimental) AAc (Experimental) AMPS (Model Prediction) AAm (Model Prediction) AAc (Model Prediction) Conversion 1 (a) Cumulative Composition Conversion (b) Cumulative Composition Conversion (c) Figure 4. Cumulative,, AMPS/AAm/AAc ( data Figure 4. Cumulative, F i, AMPS/AAm/AAc ( data model predictions) famps,0/faam,0/faac,0 = (a) 0.8/0.1/0.1, (b) 0.1/0.8/0.1 (c) 0.1/0.2/0.7. model predictions) f AMPS,0 /f AAm,0 /f AAc,0 = (a) 0.8/0.1/0.1, (b) 0.1/0.8/0.1 (c) 0.1/0.2/0.7.

15 Processes 2017, 5, Step 7: Present Reactivity Ratios JCRs It was shown Step 5 (Section.5) that usg medium-high conversion data provides smaller jot confidence regions (JCRs) estimation (compd conventional low conversion data analys). In addition, Step 6 (Section.6), estimates from medium-high conversion data were successfully used predict cumulative. Theree, estimates presented Table 4, mation last row ( ternary estimation medium-high conversion data) most prece. The JCRs associated with se estimates have been presented previously Figure. Table 4. Summary estimates AMPS 1 /AAm 2 /AAc ization. Experimental Data Conversion Type r 12 r 21 r 1 r 1 r 2 r 2 Literature [12,14] Medium-High Bary Optimally Designed Data Low Ternary Optimally Designed Data Medium-High Ternary Conclusions We have dcussed effectiveness error--variables-model (EVM) framework applications. More specifically, accurate ternary s have been establhed AMPS/AAm/AAc ization. In a comparon estimation results low conversion medium-high conversion level data, pot estimates were fairly constent. However, terms precion, medium-high conversion level data provided much smaller JCRs, which dicates a much higher degree confidence results (compd low conversion data results). Th represents an improvement, sce collected data at medium-high conversion levels conta more mation; addition, potentially accurate assumptions (required analyses with low conversion level data stantaneous models) avoided. For first time, EVM was successfully applied ization data at medium-high conversion levels. The analys has also shown that sufficient mation available from ly designed feed s; ternary s were estimated with high precion very little correlation AMPS/AAm/AAc system. Acknowledgments: The authors wh acknowledge fancial support from Natural Sciences Engeerg Research Council (NSERC) Canada Canada Research Chair (CRC) program. In addition, thanks go UWW/OMNOVA Solutions, Akron, OH, USA, special support A.J.S. Author Contributions: The data collection estimation was permed by A.J.S. The EVM framework ( design experiments parameter estimation) was based on Ph.D. s by N.K. A.P. superved work done by both N.K. A.J.S. Conflicts Interest: The authors decl no conflict terest. References 1. Ryles, R.G.; Neff, R.E. Thermally stable acrylic monomer prile modification applications. In Water-Soluble Polymers Petroleum Recovery; Stahl, G.A., Schulz, D.N., Eds.; Sprger Science & Busess Media: New York, NY, USA, Wei, B.; Romero-Zerón, L.; Rodrigue, D. Oil dplacement mechanms vcoelastic polymers enhanced oil recovery (EOR): A review. J. Pet. Explor. Prod. Technol. 2014, 4, [CrossRef]. Zaiun, A.; Makakou, P.; Bl, N.; Al-Maamari, R.; Al-Hashmi, A.; Abdel-Goad, M.; Al-Sharji, H. Shear Stability EOR Polymers. In Proceedgs Society Petroleum Engeers International Symposium, The Woodls, TX, USA, 11 1 April 2011.

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