# Optimal Design for Reactivity Ratio Estimation: A Comparison of Techniques for AMPS/Acrylamide and AMPS/Acrylic Acid Copolymerizations

^{*}

## Abstract

**:**

_{AMPS}= 0.18, r

_{AAm}= 0.85) and AMPS/AAc (r

_{AMPS}= 0.19, r

_{AAc}= 0.86).

## 1. Introduction

#### 1.1. Copolymerization Kinetics

_{1}] and [M

_{2}] are the concentrations of monomer 1 and 2 in the polymerizing mixture, and

_{1}and r

_{2}, describe the potential for homo-propagation relative to cross-propagation. These parameters are specific to each copolymer system, and many summary tables are available citing reactivity ratios of common copolymer systems [9]. Reactivity ratios can be estimated using experimental data, if the free (unreacted) monomer composition in the polymerizing mixture and the bound (incorporated) monomer composition in the polymer chains (i.e., copolymer composition) are known.

_{1}, given the comonomer composition in the polymerizing mixture.

_{1}and f

_{2}represent the mole fractions of unreacted monomer 1 and monomer 2 in the polymerizing mixture. F

_{1}is the instantaneous mole fraction of monomer 1 units bound (incorporated) in the copolymer chains, corresponding to f

_{1}.

_{1}) and the instantaneous copolymer composition (F

_{1}) are equivalent. If the reactivity ratios are known, we can use the instantaneous copolymerization equation (Equation (3)) to examine F

_{1}as a function of f

_{1}and to establish the azeotropic point. By setting F

_{1}= f

_{1}, Equation (3) is simplified to the binary azeotropic composition, shown in Equation (4) [10].

#### 1.2. Reactivity Ratio Estimation

#### 1.3. Design of Experiments

_{2,1}and f

_{2,2}denote the initial feed composition of monomer 2 for the first and second experiments, respectively. Preliminary reactivity ratio estimates (r

_{1}and r

_{2}) can be obtained from the literature or from some type of preliminary experimentation.

## 2. Experimental

#### 2.1. Reagent Purification

#### 2.2. Polymer Synthesis

#### 2.3. Polymer Characterization

## 3. AMPS/AAm Copolymer

#### 3.1. Literature Background for AMPS/AAm

_{AAm}), there are evident inconsistencies between experimental techniques and reactivity ratio estimation methods. It is also important to note that all of the estimation techniques used to date have been linear. Given the numerous sources of error associated with linear estimation methods and the advantages of non-linear techniques, it seems only reasonable that future reactivity ratios be estimated using EVM [7].

Ref. | Experimental | Estimation Technique | r_{AMPS} | r_{AAm} |
---|---|---|---|---|

[16] | --Type: Aqueous solution crosslinking copolymerization --Initiator: KPS --Temperature: 40 °C --pH = 7 --Composition: IR and EA | Comparison of feed and copolymer compositions (no statistical estimation) | 1.00 | 1.00 |

[23] | --Type: Aqueous solution copolymerization --Initiator: KPS --Temperature: 50 °C --Composition: EA | Billmeyer * [24] Billmeyer * [24] Kelen-Tudos Average | 0.76 0.70 0.62 0.70 ± 0.08 | 1.00 1.06 1.21 1.10 ± 0.10 |

[25] | --Type: Aqueous solution copolymerization --Initiator: KPS --Temperature: 35 °C and 55 °C --Composition: H-NMR and vibrational Raman spectroscopy | Fineman-Ross | 1.00 | 1.00 |

[26] | --Type: Aqueous solution copolymerization --Initiator: KPS --Temperature: 30 °C --pH = 9 --Composition: IR and EA | Fineman-Ross Kelen-Tudos Integrated Mayo-Lewis | 0.49 ± 0.02 0.52 ± 0.07 0.50 ± 0.01 | 0.98 ± 0.09 1.00 ± 0.08 1.02 ± 0.01 |

[27] | --Type: Aqueous solution copolymerization --Initiator: APS --Temperature: 60 °C --Composition: EA and C-NMR | Fineman-Ross Kelen-Tudos | 0.37 ± 0.04 0.42 ± 0.03 | 1.01 ± 0.01 1.05 ± 0.06 |

[27] | --Type: Aqueous solution redox copolymerization --Initiator: APS/NaHSO _{3} --Temperature: 25 °C --Composition: C-NMR | Fineman-Ross Kelen-Tudos | 0.54 ± 0.03 0.51 ± 0.03 | 1.07 ± 0.01 1.05 ± 0.06 |

#### 3.2. Design of Experiments for AMPS/AAm

_{AMPS}= 0.50, r

_{AAm}= 1.02) [26]. Once preliminary reactivity ratio estimates are established, the T-M and EVM criteria can be used to design optimal experiments. Each experimental design provides two feed compositions (in terms of monomer 1; AMPS in this case) at which to run new experimental trials, and the results are presented below. In Table 2, f

_{AMPS,0,1}represents the first initial feed composition (in terms of AMPS) from the design, just as f

_{AMPS,0,2}represents the second initial feed composition from the design. The reactivity ratio estimates obtained from each design are also included for easy comparison. More details on the determination of reactivity ratio estimates follow.

Approach | Reactivity Ratios for Design | Feed Compositions (Mole Fractions) | New Reactivity Ratio Estimates | |||
---|---|---|---|---|---|---|

r_{AMPS} | r_{AAm} | f_{AMPS,0,1} | f_{AMPS,0,2} | r_{AMPS} | r_{AAm} | |

Preliminary | 0.50 | 1.02 | 0.15 | 0.80 | 0.13 | 0.84 |

T-M Design | 0.13 | 0.84 | 0.30 | 0.91 | 0.16 | 0.77 |

EVM Design | 0.13 | 0.84 | 0.10 | 0.84 | 0.18 | 0.85 |

#### 3.3. Reactivity Ratio Estimation

#### 3.4. Discussion of Results

#### 3.4.1. Cumulative Copolymer Composition

_{AMPS,0}= 0.30 and f

_{AMPS,0}= 0.91), it is possible to predict the cumulative copolymer composition.

_{AMPS,0}= 0.30, for example), the model predictions are in very good agreement. In fact, the model predictions for f

_{AMPS,0}= 0.30 from the EVM-design and from McCormick and Chen [26] are almost indistinguishable. However, at f

_{AMPS,0}= 0.91, there is a significant difference in model predictions, especially when comparing the optimally-designed experiments to the literature values. The difference in prediction behavior between f

_{AMPS,0}= 0.30 and f

_{AMPS,0}= 0.91 is due to the nature of the system. When the AMPS content is low in the feed, there is very little composition drift (that is, ${\text{f}}_{\text{AMPS}}\approx {\text{F}}_{\text{AMPS}})$, which means that the reactivity ratios do not have a significant influence on the copolymer composition predictions. Conversely, when f

_{AMPS,0}is high, the propagation of error is evident in the model predictions. Again, this highlights the importance of obtaining accurate reactivity ratios in order to calculate other copolymer property trajectories properly.

**Figure 2.**Cumulative copolymer composition for AMPS/AAm; T-M-designed experiments (f

_{AMPS,0}= 0.30 and f

_{AMPS,0}= 0.91).

#### 3.4.2. Instantaneous Copolymer Composition

_{AMPS,0}= 0.84 are presented in Figure 3.

_{AMPS,0}is high), which confirms previous observations.

#### 3.4.3. Azeotrope Analysis

_{AMPS}can be plotted as a function of f

_{AMPS}to establish the azeotropic point.

**Figure 4.**Determination of azeotropic composition from (

**a**) literature data [26]; (

**b**) preliminary data; (

**c**) T-M-designed data; and (

**d**) EVM-designed data; 45° line (F

_{AMPS}= f

_{AMPS}) indicated by a dotted line.

_{AMPS}varies with f

_{AMPS}, given four sets of reactivity ratio estimates ((a) literature data; (b) preliminary data; (c) T-M-designed data; and (d) EVM-designed data). The point at which the curve passes through the 45° line (F

_{AMPS}= f

_{AMPS}, here indicated by a dotted line) represents the azeotropic composition.

_{AMPS}≈ F

_{AMPS}(that is, the F

_{AMPS}curve falls very close to the 45° line) at low values of f

_{AMPS}in all cases, which confirms the results of Figure 2. However, as expected, the curve never passes through the 45° line in case (a). From a mathematical perspective, it is only feasible to observe a non-negative azeotropic point in the binary system when both reactivity ratios are less than or greater than unity; according to McCormick and Chen [26], r

_{AMPS}= 0.50 and r

_{AAm}= 1.02. Therefore, their reactivity ratio estimates suggest that an azeotrope does not occur in this system.

_{AMPS}= 0.16 and r

_{AAm}= 0.77), the azeotrope occurs at f

_{AMPS}= F

_{AMPS}= 0.22. On the other hand, reactivity ratios from the preliminary (case (b); r

_{AMPS}= 0.13 and r

_{AAm}= 0.84) and the EVM-designed data (case (d); r

_{AMPS}= 0.18 and r

_{AAm}= 0.85) both predict the azeotropic composition to be f

_{AMPS}= F

_{AMPS}= 0.16. Hence, the location of the system azeotrope is somewhere between 0.16 and 0.22.

_{AMPS}= 0.18 and r

_{AAm}= 0.85.

## 4. AMPS/AAc Copolymer

#### 4.1. Literature Background for AMPS/AAc

Ref. | Experimental | Estimation Technique | r_{AMPS} | r_{AAc} |
---|---|---|---|---|

[28] | --Type: Aqueous solution copolymerization (<10% conversion) --Initiator: BPO --Temperature: 55 °C --Composition: IR | Fineman-Ross Kelen-Tudos Average | 0.304 0.15 0.27 | 0.915 0.98 0.95 |

[33] | --Type: Aqueous solution copolymerization (<10% conversion) --pH = 7 --Composition: EA | Fineman-Ross Behnken’s NLR | 0.194 0.187 ± 0.09 | 0.700 0.740 ± 0.13 |

_{AMPS}.

#### 4.2. Design of Experiments for AMPS/AAc

_{AMPS,0}= 0.15 and f

_{AMPS,0}= 0.80) presented unique concerns (see Table 4). At the lower AMPS feed composition (f

_{AMPS,0}= 0.15), the copolymerization was extremely slow and minimal precipitate formed. The high AMPS run (f

_{AMPS,0}= 0.80) was better in terms of conversion and copolymer precipitation, but presented other difficulties. The reaction took place very quickly, which significantly increased variability in the system. This is, to some extent, characteristic of preliminary experiments, and the error observed in the replicates decreased substantially for the optimally designed experiments.

Approach | Reactivity Ratios for Design | Feed Compositions (Mole Fractions) | New Reactivity Ratio Estimates | |||
---|---|---|---|---|---|---|

r_{AMPS} | r_{AAc} | f_{AMPS,0,1} | f_{AMPS,0,2} | r_{AMPS} | r_{AAc} | |

Preliminary | 0.27 | 0.95 | 0.15 | 0.80 | 0.48 | 0.95 |

T-M Design | 0.48 | 0.95 | 0.32 | 0.81 | 0.21 | 0.85 |

EVM Design | 0.48 | 0.95 | 0.20 | 0.73 | 0.19 | 0.86 |

_{AMPS,0}= 0.15, a constraint (0.2 < f

_{AMPS,0}< 1.0) was included when designing optimal experiments through EVM.

#### 4.3. Reactivity Ratio Estimation

#### 4.4. Discussion of Results

_{i}| is the determinant of the EVM (or T-M) design criterion for a given design of experiments. For the data of Figure 5, the JCR volume ratio is 1.1659, which indicates that the JCR from the T-M design is larger than the JCR from the EVM design [34]. The detailed calculation is provided in Appendix A, Section A.3; the analysis confirms that the EVM-designed experiments produce the smallest JCR for the AMPS/AAc copolymer. An additional advantage of the EVM-designed experiments, which is observed in both Figure 1 and Figure 5, is the decrease in correlation between reactivity ratios compared to the T-M-designed results (as indicated by the decreased slope of the error ellipse). Therefore, the EVM-designed reactivity ratios r

_{AMPS}= 0.19 and r

_{AAm}= 0.86 can be used to calculate cumulative copolymer composition profiles. Results are shown in Figure 6.

## 5. Conclusions

Copolymer | r_{1} | r_{2} |
---|---|---|

AMPS^{1}/AAm^{2} | 0.18 | 0.85 |

AMPS^{1}/AAc^{2} | 0.19 | 0.86 |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A: Experimental Data

## A.1. AMPS/AAm Copolymerization Data

Run # | X | f_{AMPS,0} | f_{AAm, 0} | ${\overline{F}}_{AMPS}$ | ${\overline{F}}_{AAm}$ |
---|---|---|---|---|---|

1 | 0.0061 | 0.30 | 0.70 | 0.3243 | 0.6757 |

0.1078 | 0.30 | 0.70 | 0.2592 | 0.7408 | |

0.2614 | 0.30 | 0.70 | 0.2683 | 0.7317 | |

0.3335 | 0.30 | 0.70 | 0.2701 | 0.7299 | |

0.4717 | 0.30 | 0.70 | 0.2841 | 0.7159 | |

2 | 0.0583 | 0.91 | 0.09 | 0.6772 | 0.3228 |

0.1483 | 0.91 | 0.09 | 0.6779 | 0.3221 | |

0.2829 | 0.91 | 0.09 | 0.7043 | 0.2957 | |

0.5207 | 0.91 | 0.09 | 0.7223 | 0.2777 | |

0.7076 | 0.91 | 0.09 | 0.7374 | 0.2626 | |

3 | 0.0671 | 0.30 | 0.70 | 0.2794 | 0.7206 |

0.1035 | 0.30 | 0.70 | 0.2626 | 0.7374 | |

0.1830 | 0.30 | 0.70 | 0.2735 | 0.7265 | |

0.2604 | 0.30 | 0.70 | 0.2797 | 0.7203 | |

0.3910 | 0.30 | 0.70 | 0.2858 | 0.7142 | |

4 | 0.0519 | 0.91 | 0.09 | 0.8335 | 0.1665 |

0.1441 | 0.91 | 0.09 | 0.7955 | 0.2045 | |

0.2710 | 0.91 | 0.09 | 0.7648 | 0.2352 | |

0.4626 | 0.91 | 0.09 | 0.7715 | 0.2285 | |

0.6151 | 0.91 | 0.09 | 0.7762 | 0.2238 |

_{AMPS,0}= mole fraction of AMPS in the initial monomer feed; ${\overline{F}}_{AMPS}$ = cumulative mole fraction (composition) of AMPS in the copolymer product. These symbols are used throughout this Appendix.

Run # | X | f_{AMPS,0} | f_{AAm, 0} | ${\overline{F}}_{AMPS}$ | ${\overline{F}}_{AAm}$ |
---|---|---|---|---|---|

1 | 0.3408 | 0.10 | 0.90 | 0.1141 | 0.8859 |

0.3425 | 0.10 | 0.90 | 0.0937 | 0.9063 | |

0.7073 | 0.10 | 0.90 | 0.0801 | 0.9199 | |

2 | 0.0731 | 0.84 | 0.16 | 0.5977 | 0.4023 |

0.1412 | 0.84 | 0.16 | 0.6332 | 0.3668 | |

0.1923 | 0.84 | 0.16 | 0.7141 | 0.2859 | |

0.3348 | 0.84 | 0.16 | 0.6555 | 0.3445 | |

3 | 0.1064 | 0.10 | 0.90 | 0.1681 | 0.8319 |

0.1473 | 0.10 | 0.90 | 0.0911 | 0.9089 | |

0.3556 | 0.10 | 0.90 | 0.0898 | 0.9102 | |

0.6174 | 0.10 | 0.90 | 0.0922 | 0.9078 | |

4 | 0.2862 | 0.84 | 0.16 | 0.7030 | 0.2970 |

0.3589 | 0.84 | 0.16 | 0.6938 | 0.3062 |

## A.2. AMPS/AAc Copolymerization Data

Run # | X | f_{AMPS,0} | f_{AAc, 0} | ${\overline{F}}_{AMPS}$ | ${\overline{F}}_{AAc}$ |
---|---|---|---|---|---|

1 | 0.0617 | 0.32 | 0.68 | 0.2259 | 0.7741 |

0.1461 | 0.32 | 0.68 | 0.2397 | 0.7603 | |

0.2613 | 0.32 | 0.68 | 0.2333 | 0.7667 | |

0.4426 | 0.32 | 0.68 | 0.2386 | 0.7614 | |

0.4426 | 0.32 | 0.68 | 0.3182 | 0.6818 | |

2 | 0.0462 | 0.81 | 0.19 | 0.6014 | 0.3986 |

0.0874 | 0.81 | 0.19 | 0.6032 | 0.3968 | |

3 | 0.0528 | 0.32 | 0.68 | 0.3701 | 0.6299 |

0.0804 | 0.32 | 0.68 | 0.3298 | 0.6702 | |

0.1177 | 0.32 | 0.68 | 0.3253 | 0.6747 | |

0.2395 | 0.32 | 0.68 | 0.3120 | 0.6880 | |

4 | 0.0524 | 0.81 | 0.19 | 0.6802 | 0.3198 |

0.1038 | 0.81 | 0.19 | 0.6849 | 0.3151 | |

0.2576 | 0.81 | 0.19 | 0.6182 | 0.3818 | |

0.2576 | 0.81 | 0.19 | 0.5992 | 0.4008 |

Run # | X | f_{AMPS,0} | f_{AAc, 0} | ${\overline{F}}_{AMPS}$ | ${\overline{F}}_{AAc}$ |
---|---|---|---|---|---|

1 | 0.0269 | 0.20 | 0.80 | 0.2652 | 0.7348 |

0.1369 | 0.20 | 0.80 | 0.2119 | 0.7881 | |

0.4156 | 0.20 | 0.80 | 0.2075 | 0.7925 | |

0.4950 | 0.20 | 0.80 | 0.1860 | 0.8140 | |

0.4950 | 0.20 | 0.80 | 0.1723 | 0.8277 | |

0.5813 | 0.20 | 0.80 | 0.1649 | 0.8351 | |

2 | 0.0895 | 0.73 | 0.27 | 0.5939 | 0.4061 |

0.1250 | 0.73 | 0.27 | 0.5115 | 0.4885 | |

0.1642 | 0.73 | 0.27 | 0.5131 | 0.4869 | |

3 | 0.1458 | 0.20 | 0.80 | 0.2474 | 0.7526 |

0.1458 | 0.20 | 0.80 | 0.1418 | 0.8582 | |

0.2951 | 0.20 | 0.80 | 0.1439 | 0.8561 | |

4 | 0.0798 | 0.73 | 0.27 | 0.6063 | 0.3937 |

0.4756 | 0.73 | 0.27 | 0.5455 | 0.4545 | |

0.5664 | 0.73 | 0.27 | 0.6069 | 0.3931 |

## A.3. Design and Joint Confidence Region Comparison Calculations for AMPS/AAc

_{i}| is the determinant of the EVM (or T-M) design criterion for a given design of experiments. If the ratio is less than unity, the EVM-designed data has a larger JCR, and the T-M design is superior. Similarly, if the ratio is greater than unity, the T-M-designed data has a larger JCR, and the EVM design is more efficient. The information used for the analysis of AMPS/AAc in Section 4.4 is provided in Table A5, below:

T-M-Designed Data: | EVM-Designed Data: |
---|---|

$\text{G}=\left[\begin{array}{cc}5369.3& -1869.2\\ -1869.2& 2661.0\end{array}\right]$ | $\text{G}=\left[\begin{array}{cc}4100.6& -1608.0\\ -1608.0& 4208.8\end{array}\right]$ |

$\left|{\text{G}}_{\text{T}-\text{MDesign}}\right|=1.0794\times {10}^{7}$ | $\left|{\text{G}}_{\text{EVMDesign}}\right|\text{}=1.4673\times {10}^{7}$ |

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**MDPI and ACS Style**

Scott, A.J.; Riahinezhad, M.; Penlidis, A.
Optimal Design for Reactivity Ratio Estimation: A Comparison of Techniques for AMPS/Acrylamide and AMPS/Acrylic Acid Copolymerizations. *Processes* **2015**, *3*, 749-768.
https://doi.org/10.3390/pr3040749

**AMA Style**

Scott AJ, Riahinezhad M, Penlidis A.
Optimal Design for Reactivity Ratio Estimation: A Comparison of Techniques for AMPS/Acrylamide and AMPS/Acrylic Acid Copolymerizations. *Processes*. 2015; 3(4):749-768.
https://doi.org/10.3390/pr3040749

**Chicago/Turabian Style**

Scott, Alison J., Marzieh Riahinezhad, and Alexander Penlidis.
2015. "Optimal Design for Reactivity Ratio Estimation: A Comparison of Techniques for AMPS/Acrylamide and AMPS/Acrylic Acid Copolymerizations" *Processes* 3, no. 4: 749-768.
https://doi.org/10.3390/pr3040749