Article pubs.acs.org/IECR
Sequence Distribution and Cumulative Composition of Gradient Latex Particles Synthesized by a Power-Feed Technique Wenxing Yang, Delong Xie, Xinxin Sheng, and Xinya Zhang* School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou, 510640, China S Supporting Information *
ABSTRACT: Nonuniform latex particles, with a gradient change in composition between molecular chains, were prepared using a power-feed technique. The feed program of the addition process was quantitatively analyzed, and the corresponding equations were derived. A simple dropwise analysis model was established to simulate the sequence distribution and cumulative composition of gradient latex particles based on traditional free radical copolymerization. This model avoided consideration of the detailed kinetic procedure with great uncertainties in an emulsion polymerization. The apparent reactivity ratio of methyl methacrylate (MMA) and n-butyl acrylate (BA) during the emulsion copolymerization process, which was first measured by utilizing K-T, YBR, and other methods, and together with the conversion function, were used as the model parameters. 1H NMR and 13C NMR were used to characterize the experimental sequence distribution and cumulative composition to verify the model. The simulation results are in good agreement with the experimental data, and the model shows great tolerance to reactivity ratios calculated by different methods. in changes in electronic and mechanical properties of final polymer product, as well as their color. In traditional free radical copolymerization (FRcoP), the composition of the instantaneously formed macromolecular chain varies constantly, mainly because of different rates of consumption for the monomers based on their reactivity ratio. That is so-called “composition drift”,33,34 which leads to the production of polymers with widely diverse structures. The spontaneity of the procedure limits applications of these polymers because of their uncertain structure.35 Our previous study shows that the combination of FRcoP with a power-feed technique would allow for precise control of the composition of the molecular chains and synthesis of polymers with desired composition profiles.36 The difference of FRcoP from CLRcoP is that the generated polymer chain would be not identical and an expectable gradient composition between molecular chains is also formed between versatile polymer chains generated at different moments. To further illustrate the undergoing process of previous work, the simulation process was conducted. There are generally two domain approaches to simulate the copolymerization process, a deterministic approach,37,38 including a set of coupled, ordinary differential equations, and a stochastic approach, which is often a Monte Carlo algorithm.39,40 Both of them require lots of kinetics parameters as input in the simulation procedure, which is sometimes hard to get and has a lack of precision. Especially when it comes to copolymerization which involves more possible reaction routines, the shortage is even worse.
1. INTRODUCTION There is a great demand of diverse and new polymers for various industrial and household applications, and the functional applications of polymer are mainly based on its unique molecular structure.1,2 Among these new polymers, gradient polymers have attracted considerable interest because of their unique structures in which the composition of the constituents has a predefined change along the copolymer chain.3,4 Gradient latex particles from two monomers A and B constitute a new class of copolymer where the polymer’s composition continuously changes along the backbone from being predominantly A to predominantly B.5−7 This particular chain structure confers the gradient latex copolymer materials with striking features as well as unique physical, mechanical, and thermal properties.8−14 Various techniques have already been explored to precisely control these chain tailor-made structures,15−28 including both innovative polymerization method, such as nitroxide mediated polymerization (NMP),15,16 atom transfer radical polymerization (ATRP),17−21 reversible addition−fragmentation chain transfer polymerization (RAFT),22−25 and single-electron transfer and single-electron transfer degenerative chain transfer living radical polymerization.26,27 More recently, the combination of controlled/living radical copolymerization (CLRcoP) with a specific polymerization program has been widely utilized to achieve that purpose and the synthesized copolymer structures are now theoretically and practically fine-tuned by combining ATRP or RAFT with the appropriate polymerization program.28−32 Since its particular “living” intrinsic of CLRcoP, the synthesized polymer chain, without chain transfer and termination, would be subject to a gradient structure in a series of identical polymer chains. However, the widely used transitionmetal catalysts limit the application in polymers, since catalysts are expensive and hard to remove from the polymers, resulting © 2013 American Chemical Society
Received: Revised: Accepted: Published: 13466
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The composition of power-feed stream is first mathematically calculated, dependent on the feed rates and monomer mass in the two tanks. Further, a dropwise kinetics analysis is established on the basis of Mayo−Lewis theory for FRcoP. The combined power-feed program and dropwise model therefore forms a completed simulation frame, where only reactivity ratio and conversion function are required as input parameters. The sequence distribution and cumulative composition are chosen as the output parameters in the dropwise model to further compare with experiment determined from 1H NMR and 13C NMR investigation. The analysis and experiments are described in detail in the following sections and subsections, respectively. Mathematical Analysis of the Power-Feed Program. As shown in Figure 2, the first important issue was to quantitatively analyze the feed-composition of gradient feed streams. An expression for the instantaneous concentration of monomer 1 (from tank 1) that enters the reactor can be developed by considering its mass equilibrium at the moment t, as shown in eq 1:
Many previous studies have focused on the power-feed technique.41−45 Bassett and Santillán41−43 reported their works on producing nonuniform emulsion particles by power-feed and established a model. The model was based on the assumption of the “starved situation” that divided the total polymerization into two processes of conversion: a supposed low value of conversion in the first stage and complete conversion in the second stage. However, the assumption does not agree with the experimental measurement results because the reaction rates gradually increase at the first stage, and it is usually difficult to carry out the reaction such that complete conversion is realized. A more reliable description of the process is therefore required. Herein, this article focuses on establishing an effective model by introducing a dropwise analysis procedure for describing the combination of the power-feed technique and traditional FRcoP. The work started with a mathematical analysis of the effect of the power-feed program on concentration profiles. When it comes to the simulation of sequence distribution and accumulative composition in the polymer chain, a simple but effective approach was proposed based on the mass equilibrium of the procedure, by avoiding tedious analysis of the kinetic process. Only three parameters (apparent reactivity ratios, feedcomposition, and conversion curve) were introduced in the model, and further experiments were conducted to verify the reliability of the model.
M[initial]1 + M[input]1 − M[output]1 = concentration1 volume (1)
At constant flow rates this equation becomes t
V10C10 − ∫ R1C1(t ) dt + R 2C20t 0 V10 + (R 2 − R1)t
2. MODEL DEVELOPMENT A typical two-tank power-feed procedure is illustrated in Figure 1. In this arrangement, two feeding tanks are connected in series
= C1(t )
(2)
where C1(t) is the concentration of the monomer 1 in tank 1 at any given time (mol/L), V01 is the volume of monomer in tank 1 (L), R1 and R2 are the feed rates (L/h), and C02 is the monomer 2 concentration in tank 2 (mol/L). Rearrangement of eq 2 and derivation yields the following expression: C1(t ) = C20 − (C20 − C10)(1 − α)x
(3)
where α = −{[(R2 − R1)t]/V01} and x = −[R2/(R2 − R1)]. Hence, eq 3 expresses the variation in the concentration of monomer 1 entering the reactor, as a function of time. However, eq 3 is only useful for describing the mixture concentration when monomers are consumed together. Other situations, when monomers are consumed in specific sequence, are summarized in Table 1. To simplify in this work, the simple situation was chosen with only methyl methacrylate (MMA) in tank 2 and n-butyl acrylate (BA) in tank 1 as a starting point. But it is not limited to that, and more practical tasks could be easily accomplished. Now, the initial concentration parameters in the simplified case are C02 = 0, C01 = 1. Thus, the equations in Table 1 become more manageable, as shown in Table 2. It can be seen from Table 2 that C1(t) is influenced by the relationships among V01, V02, R1, and R2. Examples are listed in Figure 3 to illustrate effect of different feed situations on the composition profiles. Figure 3a illustrates that when mixtures in two tanks are consumed together, the curve profiles are influenced by the value V01/V02, and the power-feed, indeed “generates” the effects of power composition change on the monomer mixture. When (R1 − R2)/R2 = 1 and x = 1, the concentration function is equal to C1(t) = (1 − α), which is a linear relationship with t. On the other hand, for x > 1 and x < 1, the relation is not linear and shows different profile curvature.
Figure 1. Schematic illustration of two-tank power-feed polymerization.
to a reactor. The monomer mixture in tank 2 is continuously fed to the monomer mixture in the well-stirred tank 1. The continuously changing mixture in tank 1 is then simultaneously fed into the reaction vessel in a manner typical for emulsion copolymerization. In order to develop a mathematical model, three assumptions were made: (i) the feeding flow to the reactor does not change during synthesis, (ii) the feeding tanks are empty at the same time, and (iii) the agitation in feeding tanks is sufficient to homogenize the pre-emulsions. When the feed technique shown above is utilized to take control of the reaction, the total procedure can generally be divided into two processes: the first is generation of gradient feed streams with continuous composition changed by the power-feed technique, and the second is a batch FRcoP, where mixing and reaction take place simultaneously. The design procedure of the dropwise analysis model is shown in Figure 2. 13467
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Figure 2. Design procedure of the drop-analysis model.
Table 1. Concentration Function under Variable Feed Rates case
tmix and tall
consume together
tmix = tall = V02/R2 = (V01+V02)/R1
c1(t ) = c 20 − (c 20 − c10)(1 − α)x
tank 1 first
R1tmix = tmixR2 + V01 tall = V02/R2
0 0 0 x ⎧ t < tmix ⎪ c 2 − (c 2 − c1 )(1 − α) c1(t ) = ⎨ ⎪ 0 c tmix ≤ t ≤ tall ⎩ 2
tank 2 first
tmix = V02/R2 tall = (V01 + V02)/R2
0 0 x ⎧ 0 t < tmix ⎪ c 2 − (c 2 − c1 )(1 − α) c1(t ) = ⎨ ⎪ tmix ≤ t ≤ tall ⎩ c1(tmix )
concentration function
Table 2. Concentration Function under C02 = 0, C01 = 1 case
tmix and tall
consume together tank 1 first
tmix = tall = V02/R2 = (V01 + V02)/R1 R1tmix = tmixR2 + V01 tall = V02/R2
tank 2 first
tmix = V02/R2 tall = (V01 + V02)/R2
Dropwise Analysis Procedure. In the above-described power-feed program, the reactants are fed into the reactor in a dropwise manner. At all times, the emulsion polymerization actually proceeds in a manner similar to that in a batch reactor. Therefore, with the instantaneous feeding composition, the reaction procedure can be divided into numerous batch reactions in sequence, as illustrated in Figure 4. The feeding monomers with composition f fed,n first mix together with the residual fend,n−1 monomers left in a previous n − 1 moment, and then, FRCoP takes place with certain conversion. The mole fraction of accumulative unreacted monomer is then fend,n. Repeatability of the procedure of mixture and reaction propagates until the end of reaction. The iterative calculation of the above batch procedure is defined here as dropwise analysis. Sequence distribution and cumulative composition can be simulated using this model. Copolymer Composition Based on Dropwise Analysis. The elementary reactions in the power-feed FRcoP of monomer M1 with monomer M2, simulation of sequence distribution, and cumulative composition is illustrated as Table 3. Assumption of the procedure was made according to the Mayo−Lewis copolymerization model47 where no chain transfer reaction and
concentration function
c1(t ) = (1 − α)x x ⎧ t < tmix ⎪(1 − α) c1(t ) = ⎨ ⎪ 0 tmix ≤ t ≤ tall ⎩c2 x ⎧ t < tmix ⎪(1 − α) c1(t ) = ⎨ ⎪ c ( t ) tmix ≤ t ≤ tall ⎩ 1 mix
Figure 3b illustrates that besides the different curvature caused by V 01/V 02, the consumption order of the tanks also has an effect on the feed composition profiles, where monomer mass actually matters. After the consumption of one monomer, the residual concentration is different, depending on in which tank monomer finished first. A similar mathematical analysis process could be used for three or more tanks, which produce more complicated composition changes, giving extreme freedom to create a desired feed composition profile. For example, a three-tank arrangement leads to inflections and composition reversals.46 13468
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Table 3. Elementary Reactions Involved in FRcoP Reaction chain initiation
k i1
R· + M1 → RM1· k i2
R· + M 2 → RM 2· chain propagation
k11
∼ M1· + M1 → ∼ M1· k12
∼ M1· + M 2 ⎯→ ⎯ ∼ M 2· k 21
∼ M 2· + M1 ⎯→ ⎯ ∼ M1· k 22
∼ M 2· + M 2 ⎯→ ⎯ ∼ M 2· chain termination
k t11
∼ M1· + ·M1 ∼ ⎯⎯⎯→ ∼ M1M1∼ k t12
∼ M1· + ·M 2 ∼ ⎯⎯⎯→ ∼ M1M 2∼ k t22
∼ M 2· + ·M 2 ∼ ⎯⎯⎯→ ∼ M 2M 2∼
is no mixture with another monomer, f1̅ , the average mole fraction during the batch reaction is approximately equal to f1/2. Polymerization procedure undertakes with a conversion, m1. By the time when the second drop was fed to the system, the amount of monomer consumed in the first drop is [M]fed m1, thus there is still residual [M]fedm1(1 − m1) unreacted monomer in the system. The produced polymer F1 can be calculated according to eq 4, and the residual monomer composition can be obtained using the following equation: Figure 3. Concentration curve under different situations. (a) Concentration curve when monomers are consumed together. (b) Concentration curve when monomers are consumed in order.
fend,1 =
r1f12 + f1 f2 r1f12 + 2f1 f2 + r2f2 2
[M]fed (1 − m1)
=
ffed,1 − m1F1̅ 1 − m1
(5)
Until now, both the residual monomer and generated polymer composition has not been available. The detailed reduction relation involved is summarized in Table S1. When the second drop with composition f fed,2 is introduced into the system, it would first mix together with the residual monomer fend,1 from the first drop generating a mixture monomer composition, f2̅ . Then, the reaction would take place with a conversion of s2, the conversion between the second and third drop, before the feed of the third drop. It is worthy to notice that s2 is referred to the instantaneous conversion between drops, while another variables m2 which is also widely used in our calculation are defined to be the entire monomer conversion until the end of specific drops, shown in the following formula. Similar to the first drop analysis, eq 4 is utilized
pseudo-steady-state were supposed. The relationship between the instantaneous copolymer composition, F1, and monomer composition, f1, can be expressed by the following equation:
F1
ffed,1 [M]fed − [M]fed m1F1
(4)
where f1 = [M]1/([M]1 + [M]2) and F1 = d[M1]/(d[M1] + d[M2]). Now combination between dropwise analysis with the Mayo−Lewis model is made in the following statement. First of all, the first drop is introduced into the reaction tank, whose mole fraction is f1 and monomer mass is [M]fed. Because there
Figure 4. Illustration of dropwise analysis procedure. 13469
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again to get the polymer composition. Actually, the repeat analysis can be easily generalized to nth drop. Details are included in Supporting Information Table S1. total polymer product during two drops m2 = total added monomer during two drops
Table 4. Reactivity Ratio Experimental Results
(6)
An iterative program here can be used for delivery information between drops. There are only four parameters needed to solve this model: r1, r2, f fed,n, and the conversion function. Sequence Distribution Based on Dropwise Analysis. Similarly, it is very easy to statistically derive the sequence distribution by the dropwise model on the basis of the firstorder Markov model.48 It is assumed that the reactivity of the propagating species is independent of the identity of the monomer unit, which precedes the terminal unit. Then, the transition or conditional probability P11 of forming an M1M1 dyad in the copolymer chain can be given by the ratio of the rate for M*1 by adding M2 to the sum of the rates for M1* adding M1 and M2, that is P12 =
run
X
Y
conversion, %
1 2 3 4 5 6 7 8
9.0000 4.0000 2.3333 1.5000 1.0000 0.6667 0.4286 0.2500
24.1933 9.9200 6.6267 3.8000 2.9000 1.7933 1.1867 0.6733
2.95 4.01 4.11 7.23 6.01 1.21 2.07 5.26
Table 5. Summarization of Reactivity Ratios by Different Methods method
r1
r2
R2
r1r2
Kelen−Tüdõs extended Kelen−Tüdõs YBR method nonlinear regression
2.62 2.65 2.63 2.71
0.34 0.33 0.35 0.36
0.9862 0.9859
0.8908 0.8745 0.9205 0.9756
0.9982
k12[M1*][M 2] 1 = = 1 − P11 1 + r1X k11[M1*][M1] + k12[M1*][M 2] (7)
Similary P21 =
k 21[M*2 ][M1] 1 = 1+ k 22[M*2 ][M 2] + k 21[M*2 ][M1]
r2 X
= 1 − P22 (8)
where X = [M1]/[M2]. The transition probability is a function of function of r1, r2, and X. Because X, which is the mole fraction, changes across the reaction, the transition is not a constant, and thus, it is different between each drop. Therefore, the sequence distribution probability described also varies at different reaction moment. The quantitative value can be calculated by replacing X with f ̅, the mole fraction derived in the dropwise analysis of cumulative composition. With the sequence distribution probability, the formed triad sequence amount for each drop, (CP3{M1M1M1}, CP3{M2M1M1}, CP3{M2M1M2}, CP3{M2M2M2}, CP3{M1M2M2}, CP3{M1M2M1}) is then equal to forming probability multiply reacted monomer mole, ([M]add + [M]fed,n−1) × Sn. Hence, the cumulative triads sequence of corresponding composition, AP3{M}, is listed in Supporting Information Table S2. The parameters needed for sequence distribution are the same as those for cumulative composition. While f fed,n has been derived from mathematical analysis of the model, experiments were conducted to determine r1, r2, and the conversion function, which are to be described in the next section.
Figure 5. Comparison of the copolymerization curve between MMA/ BA and the ideal situation.
3. EXPERIMENTS Materials. Methyl methacrylate (MMA) and n-butyl acrylate (BA) (99%, Guangzhou Chaoyun Chemical Co) were purified by reduced pressure distillation (under 300 Pa and 100 °C) to remove the inhibitors. The initiator potassium persulfate (KPS, K2S2O8, 99%, Chengdu Kelong Chemical Co.) was recrystallized before use. The anionic surfactant dodecyl diphenyl ether sodium disulfonate (DSB, 99%, Chengdu Kelong Chemical Co.) was used as received without further purification. Deionized water (self-made, specific conductance
