Statistical design strategies to optimize properties in miniemulsion polymerization of methyl methacrylate

European Polymer Journal 45 (2009) 1208–1216

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European Polymer Journal journal homepage: www.elsevier.com/locate/europolj

Statistical design strategies to optimize properties in miniemulsion polymerization of methyl methacrylate G. Kermabon-Avon, C. Bressy *, A. Margaillan Laboratoire MAtériaux Polymères, Interfaces, Environnement Marin (MAPIEM), Institut des Sciences de l’ingenieur de Toulon et du Var, EA4323, Av Georges Pompidou, 83162 La Valette du Var, France

a r t i c l e

i n f o

Article history: Received 25 July 2008 Received in revised form 15 December 2008 Accepted 22 December 2008 Available online 30 December 2008

Keywords: Miniemulsion Methyl methacrylate Factorial design of experiment Particle size Latex

a b s t r a c t PMMA particles were synthesized by a miniemulsion polymerization method using hexadecane as costabilizer and sodium dodecyl sulfate as surfactant. Full factorial experimental design incorporating the linear regression analysis of the experimental values was used to illustrate the usefulness of this technique in miniemulsion polymerization studies. The effect of initiator concentration, costabilizer concentration, surfactant concentration, sonication time and amplitude, and their interactions on the particle size were identified. Costabilizer and surfactant concentrations influence both particle size individually whereas initiator concentration influences particle size via its interactions with the costabilizer and the surfactant. Sonication parameters influence also greatly the particle size. Two mathematical models were established, and have demonstrated the capability of predicting particle diameter from the synthesis conditions with a precision of 1–2 nm over a range from 75 to 180 nm. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In recent years an increasing interest is observed in the development of environment-friendly paints and coatings. This trend has been spurred by a growing concern for environmental issues, such as volatile organic solvent emissions and recycling or waste disposal problems [1,2]. This fact has forced the paint industry to develop new alternative formulations [3], such as new water-based paint formulations and high solid systems with low Volatile Organic Compounds (VOCs) [4,5]. Great stability for these latexes should be expected and for this reason the preparation of emulsion via miniemulsion polymerization has been selected. Miniemulsions are classically defined as aqueous dispersions of relatively stable oil droplets within a size range of 50–500 nm prepared by shearing a system containing oil, water, a surfactant, and an osmotic pressure agent [6]. For heterophase polymerization, nucleation of particles comes from three * Corresponding author: Tel.: +33 494 142 580; fax: +33 494 142 448. E-mail address: [email protected] (C. Bressy). 0014-3057/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eurpolymj.2008.12.033

mechanisms: micellar nucleation, homogenous nucleation and droplet nucleation [7]. Miniemulsion is known to be governed by droplet nucleation. To prevent micellar nucleation, the aqueous phase concentration of the surfactant should be below the critical micellar concentration (CMC) after shearing even if, the overall surfactant concentration may be used above or below the CMC [8]. Several types of costabilizer acting as osmotic pressure agent can be selected to suppress degradation by Ostwald ripening [6]. Initiator can be either oil- or water-soluble. In the case of an organo-soluble initiator, it is dissolved in the monomeric phase prior to the miniemulsification stage. Then the reaction starts within the droplets. In the case of a water-soluble initiator, polymerization starts from the continuous aqueous phase, similarly to conventional emulsion polymerization where water-soluble initiators are used usually as they contribute to the charged character of the latex particle [9]. Understanding of the miniemulsion polymerization is required as a mean of reproducibility for the formulation of waterborne paints. Several authors [6,10–17] have already studied different parameters of the miniemulsion such as surfactant, initiator or

G. Kermabon-Avon et al. / European Polymer Journal 45 (2009) 1208–1216

costabilizer amount and type, sonication parameters without studying the link between these factors. The aim of this work was to study the effect of the values of different parameters on particle size that characterizes the polymer obtained by miniemulsion polymerization of methyl methacrylate. A factorial experimental design was used to elucidate the main trends and interactions between variables illustrating the application of these techniques to polymerization studies. Factorial experimental design techniques are extremely useful while using a minimum number of experiments. A model correlating the experimental conditions and the properties of the polymers by a response surface will be useful to prepare polymer with tailor-made properties.

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2. Experimental part

estimated from a calibration curve relative to PMMA standards. For sample preparation, 10 mg of the dried latex was dissolved in 1 ml of THF and filtered through a 0.45 lm Millipore filter. The average particle size and morphology were determined by Transmission Electron Microscopy (Philips EM 400 electron microscope operating at 200 kV). For sample preparation, the original latex was highly diluted with deionized water until a hardly visible turbidity was reached. Five drops of the diluted sample was placed on a 400 mesh carbon-coated copper grid and left to dry at room temperature. As PMMA usually degrades in the electron beam, particles were protected by a carbon film. Liquid 1H NMR spectra were recorded with a Bruker Avance 400 using deuterated acetone as solvent. Percentage solid content (%SC) of latex was calculating using expression (Eq. (1)):

2.1. Chemicals

%SC ¼

Methyl methacrylate (MMA purchased from Acros) was freshly distilled under reduced pressure and stored at 4 °C. 2,2’-azobis(2-methylpropionitrile) as initiator (AIBN purchased from Fluka) was recrystallized from ethanol and stored at 4 °C. Hexadecane (HD, from Acros), dodecane (DD, from Aldrich) as costabilizers and sodium dodecyl sulfate (SDS, from Fluka) as surfactant were used as received.

where m is the weight of the dried latex and m0 is the initial weight of latex placed in the petri dishes.

2.2. Synthesis of latex particles Twenty-five grams of the monomer (0.250 mol), costabilizer and initiator were stirred for 15 min at room temperature and added to a solution of SDS in 81 ml of deionized water previously mixed for 15 min. After stirring for 15 min at room temperature, the miniemulsion was prepared by ultrasonicating the pre-emulsion for different times at fixed amplitude (Sonics Vibracell 750W) at 0 °C in order to prevent the polymerization. Sonication may increase temperature up to 30 °C. Then, the mixture was transferred in a three-neck round bottom flask equipped with a cooler and a gas inlet, bubbled with nitrogen for 20 min, and the temperature was raised to 70 °C within 5 min. Solid content was close to the value of 23.6% wt. Completion of the reaction was observed after 2 h, as checked by NMR spectrometry by the absence of the 1H NMR peaks assigned to the vinylic protons of the monomer at 5.60 and 6.02 ppm. 2.3. Analysis The particle sizes (intensity average values) were measured using a Zetananosizer (model S, Malvern Instrument) at a fixed scattering angle of 90°. For sample preparation the original latex was diluted to a 2wt.% solution in deionized water. The molecular weights of PMMA samples were determined by Size Exclusion Chromatography analysis performed on a Waters 1515 isocratic HPLC pump with a Waters 2414 RI detector and 4 Styragel columns (HR5, HR4, HR3, HR1 from polymer laboratories) in THF with a flow rate of 1 ml/min at 30 °C. The molecular weights were

m  100 m0

ð1Þ

2.4. Factorial design of experiments (DOE) A factorial design of experiments was used in planning experiments that study the effects of five main factors of the miniemulsion polymerization on particle size: (X1) initiator (AIBN) concentration, (X2) costabilizer (hexadecane) concentration, (X3) surfactant (SDS) concentration in a first time, and (X4) sonication time, (X5) sonication amplitude in a second time. Fixed and design levels for the five factors are given in Table 1. The five factors have been the object of two factorial designs of experiments for which the domain are represented in Fig. 1. The experiments were run following the design matrix built in Tables 2 and 3.

3. Results and discussion 3.1. General characteristics of miniemulsion polymerization 3.1.1. Spherical particles Table 4 shows that the particle size of the latex varies over a range of 80–160 nm with a low polydispersity (generally r < 0.1). The diameter that is measured in Dynamic Light Scattering (DLS) is called the hydrodynamic diameter and refers to how a particle diffuses within a fluid [18]. The diameter obtained by this technique is that of a sphere that has the same translational diffusion coefficient as the particle being measured (Eq. (2)).

dðHÞ ¼

kT 3pgD

ð2Þ

With d(H) = hydrodynamic diameter (m), k = Boltzmann’s constant (J K1), T = absolute temperature (K), g = viscosity (106 Pa s), and D = diffusion coefficient (m2 s1). The translational diffusion coefficient will depend not only on the size of the particle ‘‘core”, but also on any surface

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Table 1 Factors and levels for factorial experimental design. Factors

Low level ()

X1 X2 X3 X4 X5

AIBN HD SDS Sonication time Sonication power

High level (+)

Real

Coded

Real

Coded

0.031 mol Laq1 0.056 mol Laq1 0.015 mol Laq1 5 min 21%

1 1 1 1 1

0.154 mol Laq1 0.308 mol Laq1 0.031 mol Laq1 25 min 90%

1 1 1 1 1

Sonication Time (Factor X4)

AIBN (Factor X1)

SDS (Factor X3) Sonication Power (Factor X5)

HD (Factor X2)

Fig. 1. Domains of study for factor X1, X2, X3 and then X4, X5.

Table 2 Design matrix of experiments for factor X1, X2 and X3. Experiments No.

Run order

GK1-160 GK2-008 GK2-010 GK2-012 GK2-054 GK2-016 GK2-058 GK2-014

1 2 3 4 5 6 7 8

X1

X2

X3

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

Table 3 Design matrix for factors X4 and X5. Experiments No.

Run order

KG01-10 KG02-11 KG03-14 KG04-16

1 2 3 4

X4

X5

1 1 1 1

1 1 1 1

structure, as well as the concentration and type of ions in the medium. This means that the size can be larger than measured by transmission electron microscopy as seen in Fig. 2, where the particle is removed from its native environment. Even with the carbon film to prevent particles from degradation in the electron beam, some may still exist and explains the smaller size observed by TEM. All size data in this article are measured by DLS. 3.1.2. Droplet nucleation Generally for polymerization by miniemulsion, the predominant mechanism is droplet nucleation. The overall concentration based on the water phase might be well above the CMC, but by subdividing the monomer into submicron droplets, a large surface area for surfactant adsorption is generated which can reduce the aqueous phase concentration to levels well below the CMC. Mouran et al. [19] reported that for SDS, in water saturated with MMA, this value is 0.0062 mol Laq1 at 25 °C. They mentioned that using SDS at 0.02 mol Laq1 for droplet sizes varying from 130 to 200 nm leads to concentration above the CMC after sonication. Fewer amounts of SDS

Table 4 Size and number of droplets and particles in the MMA miniemulsion polymerizations. Experiments

DLS data Before polymerization (Dd)

GK2-008 GK2-010 GK2-012 GK2-014 GK2-016 GK1-162c

Dp/Dd

Droplets number (Ndb)

Particles number (Npb)

Np/Nd

0.91 0.91 0.93 0.89 0.84 1.04

4.49E+17 2.42E+17 2.57E+17 3.18E+17 5.52E+17 1.39E+17

4.27E+17 2.14E+17 2.40E+17 4.05E+17 6.09E+17 1.31E+17

0.95 0.89 0.93 1.27 1.10 0.94

After polymerization (Dp)

Dz (nm)

Polydispersity (ra)

Dz (nm)

Polydispersity (ra)

102 126 123 115 95 151

0.064 0.193 0.071 0.196 0.088 0.163

93 114 115 102 80 157

0.595 0.098 0.080 0.104 0.075 0.162

a r, given by the DLS apparatus , characterizes the particle size distribution (0 < r < 1); it corresponds to the ratio of the variance over the square of the average particle diameter. The particle size distribution is generally considered as narrow when r is below 0.100. b Np was calculated using Eq. (3). c Experiment out of bounds of the factorial design of experiments with AIBN – 0.031 mol Laq1, SDS – 0.031 mol Laq1 and Hexadecane – 0.771 mol Laq1.

G. Kermabon-Avon et al. / European Polymer Journal 45 (2009) 1208–1216

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Fig. 2. Comparison of size measurement by DLS (a) and TEM (b).

than 0.02 mol Laq1 in the aqueous phase lead to concentration below CMC after sonication. Therefore, experiments GK2-008, GK2-010, GK2-012, in Table 4, have SDS concentration below the CMC. The remaining experiments have concentration above the CMC. Table 4 shows that the ratio of the particle size to the droplet size (Dp/Dd) is between 0.84 and 1.04, and the ratio of the particles number to the droplets number (Np/Nd) is between 0.89 and 1.27. Reimers and Schork [20] demonstrated that ratio Dp/Dd (Particle size/Droplet size) and ratio Np/Nd (Particles number/Droplets number) close to 1 explained primarily droplet nucleation. When the ratio is around unity, every droplet which is nucleated leads to one particle. A value lower than unity would imply incomplete droplet nucleation and a higher value would indicate the influence of micellar nucleation. In the case of experiments GK2-014 and GK2-016, Np/Nd is above one which means that micellar nucleation is also present. This result is consistent with the fact that for these experiments SDS has been used in amount above the CMC. However GK2162, which has also SDS concentration above CMC does not show micellar nucleation with value of Np/Nd below unity. It does not mean that no micelles are present in the aqueous phase. The fact that this experiment has a high concentration of costabilizer may explain this result. For amount of surfactant below the CMC value, the majority of the droplets have been nucleated to form polymer particles assuming that no homogenous nucleation occurs. For hydrophilic monomers such as methyl methacrylate, this type of nucleation could be avoided using oil soluble initiators.

3.1.3. Role of the costabilizer and its efficiency As explains by Landfester [21], the use of a costabilizer agent is necessary to suppress Ostwald ripening. It allows the build up of an osmotic pressure in the droplets which limits their shrinkage or growth and thus stabilizes the miniemulsions. Since the size and stability of the droplets depend mainly on the nature of the costabilizer used [6], two miniemulsions were prepared using different costabilizers. As predicted [21] the latexes synthesized from dodecane are bigger in size at a given concentration. As shown in Table 5, hexadecane gives the best stability to the droplets. Since water solubility of dodecane is greater than for hexadecane [22], this latter stabilizes more efficiently the droplets against Ostwald ripening. Therefore, smaller particles are obtained after polymerization considering miniemulsion as a 1:1 technique. The shelf stability of these latexes, defined as the period where no degradation by sedimentation is observed, varies from few months to more than a year which makes them suitable for paints and seems to be related to the particle size and their distribution. The smaller are these parameters, the longer is the shelf stability. The amount of costabilizer has an influence on the size of the particle as it remains in it after polymerization. Usually a minimal concentration of 0.005 mol Laq1 of costabilizer is necessary to suppress Ostwald ripening. It is not unusual to see variation in that concentration up to 0.136 mol Laq1. Landfester et al. [13] observed nanocapsules of oil encapsulated particles using costabilizer concentration varying from 0.146 to 0.442 mol Laq1 with methyl methacrylate miniemulsion polymerization. These

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Table 5 Latex characteristics with the use of different hydrophobes for SDS – 0.031 mol Laq1 and AIBN – 0.031 mol Laq1. Experiments

Hydrophobe

GK1-041 GK1-130 GK1-132 GK2-058 GK1-162 GK1-164

DLS data Amount (mol Laq1)

Dz (nm)

Polydispersity (r)

Dodecane Dodecane Hexadecane Hexadecane Hexadecane Hexadecane

0.056 0.41 0.056 0.308 0.771 1.264

111 177 75 114 157 152

0.107 0.183 0.070 0.163 0.162 0.130

high concentrations may act as coalescing aid for film formation [23]. PMMA is regarded as rather polar (but is not fully water soluble), whereas hexadecane is very unpolar so that the spreading coefficients are of the right order to stabilize a structure in which a hexadecane droplet core is encapsulated by a PMMA shell surrounded by water [13]. Therefore high concentrations of costabilizer give bigger size of particles. It seems however, that above a certain amount, the size of the particle reaches a limit as depicted in Fig. 3. It may be attributed to the transition from hexadecane acting as an osmotic pressure agent to hexadecane acting as oil encapsulated in particle. For the same experiments, the particle number decreases until a limit which is consistent with the fact that Np is related to the particle size by (Eq. (3)):

Np ¼

6s pqp Dz3

ð3Þ

Where s is the solid content, qp is the polymer density and Dz the average diameter given by DLS. In that case, increasing hexadecane content means a decrease of the shell thickness as the polymeric phase is kept constant. This limit is explained by the equilibrium between costabilizer and polymeric phase ratio. More costabilizer will lead to phase separation with spreading coefficients resulting to non-engulfed structure [13]. 3.2. Analysis of the effects through factorial design of experiments Factorial design was used to build a quantitative correlation model between the properties of the polymeric particles and the synthesis conditions. The number of factors able to affect the characteristics of the final polymeric par-

17 16 17 17 17 17

6 1 1 4 4 5

months month year months months months

P  Nj¼1 yj ; N N : number of experiment: j¼1 yjþ

ð4Þ

P where yj+ is the sum of responses yj with factor Xi at level P +1 (high level). Respectively, yj- is the sum of yj responses with the factor Xi at level 1 (low level). A simple method to determine if a factor has a significant effect is to estimate the measurement error DE and

1.0E+18

150

8.0E+17

50

+ + + + + +

PN

Effect : Ei ¼

200

100

2.75E 9.30E 7.85E 2.17E 1.31E 1.45E

Shelf stability

ticles is very large as seen in Fig. 4 with a fishbone diagram inspired from the Ishikawa cause and effect diagram [24]. The number of experiments necessary would be too high, increasing the difficulty of the analysis of the results. In consequence, taking into account our preliminary experiments and the information found in the literature [6,10– 17], the influence of certain factors was not studied, and their values were fixed according to the available information. For example, the upper limit of the costabilizer was chosen at 0.308 mol Laq1 since it does not have any influence above this value as demonstrated in Fig. 3. Once the different factors and responses had been identified, the next step was to run the experiments and analyze the results. The values of all responses for all experimental runs carried out are summarized in Tables 6 and 7, which also indicates the levels of the factors in each experiment. The values of the levels assigned to factor X1X2 are those of the interaction among the two factors X1 and X2, i.e., each level assigned to factor X1X2 is obtained by multiplying the values of columns X1 and X2 in each row. The values for columns X1X3, X2X3 and X1X2X3 are calculated in a similar manner. M is coded 1 and stands for the arithmetic mean of all experiment results. From the values of the responses, the effects of each factor Xi or interaction were calculated as follows:

Np

Particle size (nm)

Np

Type

6.0E+17 4.0E+17 2.0E+17

0

0.0E+00 0

0.5

1

Hexadecane (mol.L aq -1)

1.5

0

0.5

1

Hexadecane (mol.L aq -1)

Fig. 3. Influence of the amount of costabilizer (hexadecane – mol Laq1) on particle size and Np.

1.5

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RAW MATERIALS

PROCESSES T°C

Geometry

Reactor Stirring

Surfacant

Deionized Water

Power

Initiator

Sonication

Geometry

Time

Blade

Monomer

Hydrophobe

Material

TEM

Pression

Balance

T°C

Motor GPC

DLS

Particle Size Mw

Deionized water Stirring blade

MEASUREMENTS

Gaz

PROCESSES

REACTION MEDIUM

Fig. 4. Factors able to affect the polymeric particles obtained by miniemulsion polymerization.

compare it to the calculated effect of the factor E as follows:

Table 7 Design matrix and values of the responses for factors X4 and X5. Experiments

E >> DE: the factor has a significant effect. E << DE: the factor hasn’t a significant effect E  DE: the factor hasn’t a significant effect or very lightly First of all, the arithmetic mean (Eq. (5)) of these experiments is determined and then from this value the standard deviation (Eq. (6)) and the variance (Eq. (8)) are calculated. The measurement error (Eq. (9)) is then calculated from repeated experiments. Here run 5 has been repeated three times as seen in Table 8.

1 ðy þ y2 þ y3 Þ 4 1 i¼n i¼n i¼n X X X Standard deviation : ðyi  yÞ ¼ yi  y Arithmetic mean : y ¼

i¼1

i¼1

Standard deviation square :

ð5Þ

Run order

KG01-10 KG02-11 KG03-14 KG04-16

1 2 3 4

X4

X5

X4X5

M

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

Response Particle size Dz (nm) 179 104 112 93

Table 8 Determination of the measurement error. Experiments No.

Run order

GK2-054 GK1-132 GK1-166

5 9 10

ð6Þ

X1

X2

X3

1 1 1 DE

1 1 1

1 1 1

Responses Particle size Dz (nm) 76 75 80 ±1

i¼1

i¼n X ðyi  yÞ2

ð7Þ

i¼1

Variance : s2 ¼

No.

i¼n 1X ðy  yÞ; n i¼1 i

n : number of experiment pffiffiffiffiffi s2

ð8Þ

Measurement error : DE ¼

ð9Þ

Fig. 5 shows the plot of the effects of the factors on the size response and their significance versus the measurement error (DE). The effects or interactions which have an influence are those for which the values are above or under the ±DE value. The specific variables that influence the particle size are identified by application of statistical design. The sonication time and amplitude, the costabilizer

Table 6 Design matrix and values of the responses for factors X1, X2 and X3. Experiments No.

Run order

GK1-160 GK2-008 GK2-010 GK2-012 GK2-054 GK2-016 GK2-058 GK2-014

1 2 3 4 5 6 7 8

X1

X2

X3

X1X2

X1X3

X2X3

X1X2X3

M

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

Responses Particle size Dz (nm) 76 93 114 115 76 80 114 102

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20

20

X2

15

X4X5

15 10 5 Effect

0

10

+Δ E + -Δ E

-5 -10

Effect

-15 -20

5

X5

-25 X4 -30

X1

+ E +Δ

X2X3

0 X1X2X3

- E -Δ X3

X1X3 X1X2

-5

-10 Fig. 5. Effect of the factors and interactions on the particle size.

HD and the surfactant SDS are found to be the influencing factors on the particle size of the latex. The sonication amplitude and time influence in the negative sense the particle size. This result could be explained by the amount of cumulated energy supplied to the system which increases with increasing the intensity of these two factors. The longer the sonication is applied the smaller are the particles. The same observation can be made with the sonication amplitude. The positive influence of the costabilizer is predictable as the localization of the costabilizer is within the particle, and so a rise of its concentration gives an increase of the particle size as seen previously. The negative influence of the surfactant on the particle size is well known [6]: the more the concentration is important the more the coverage of small particle can be achieved. As seen in Fig. 5, the factors having an effect on the size of the particle response seem to be X2 (costabilizer concentration), X3 (surfactant concentration). Their interaction (X2X3) is not significant but can be explained as a null effect of their opposite effect on the particle size.

a

-1.5

b

X1X2 120

-1

X1 (initiator concentration) has no or little effect on particle size. Although X1 and X2 factors have, respectively, no and a positive effect on the particle size, interactions X1X2 (interaction between initiator and costabilizer) has a significant effect. To understand this result one need to consider the plot of their interaction represented in Fig. 6a. When X2 is at the low level, the influence of factor X1 over X2 becomes important and thus limits the costabilizer stabilization effect. The stabilization is given by the costabilizer and depends mainly from its low water solubility. Using oil-initiator may influence stabilization as they can act by themselves as costabilizers [25]. Alduncin et al. showed that the 2,2’-azobis(2-methylpropionitrile (AIBN) is not water insoluble enough to avoid the Ostwald ripening effect compared to Benzoyl peroxide (BPO) and Lauryl peroxide (LPO). The water solubility of AIBN (0.04 g/100 g H2O) and hexadecane (3.98  109 g/ 100 g H2O) mixed together as one costabilizer is higher than the one of hexadecane alone, which results in a less efficient suppression of Ostwald ripening and so gives rise to bigger particle size. When X2 is at the high level, the influence of X1 over X2 is not important and is

c

X1X3 120

X2X3 120

110

110

110

100

100

100

90

90

90

80

80

80

70

70

-0.5

0

X2 - level (-1)

0.5

1

X2 - level (+1)

1.5 -1.5

-1

-0.5 X3 - level (-1)

70 0

0.5

1

X3 - level (+1)

1.5 -1.5

-1

-0.5

0

X3 - level (-1)

Fig. 6. Diagram of the interactions X1X2 (a); X1X3 (b) and X2X3 (c) for the particle size response.

0.5

1

X3 - level (+1)

1.5

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3.3. Mathematical models

X4X5 170

Following the principles of factorial design, the particle size can be mathematically modeled as follows: (Eqs. (10) and (11)):

150 130 110

Y1 ¼ M 0 þ E1 X1 þ E2 X2 þ E3 X3 þ E12 X1X2

90

þ E13 X1X3 þ E23 X2X3 þ E123 X1X2X3

70 -1.5

-1

-0.5

0

0.5

X5 - level (-1)

1

ð10Þ

Y2 ¼ M 0 þ E4 X4 þ E5 X5 þ E45 X4X5

1.5

X5 - level (+1)

Fig. 7. Diagram of the interactions X4X5 for the particle size response.

under the limit of interpretation of the DE. When X3 is at the low level (Fig. 6b), factor X1 gives rise to an increase of the particle size. However, when X3 is at the high level, the effect of X1 with X3 is under the limit of interpretation of the measurement error. Fig. 6c shows similar slope for X2 when X3 is, respectively, at the low and high level. It means that there is no interaction between X2 and X3. As seen in Fig. 7, the slope of X4 (sonication time) when X5 (sonication amplitude) is at the low level is greater than when X5 is at the high level. This means that the factor sonication time at high amplitude of sonication is not as influent as it is alone. The steady state for the droplet size is attained quicker at high amplitude of sonication.

ð11Þ

where Y represents the value of the particle size. M0 is the mean of all responses, E1, E2 and E3 represent the principal effect associated with each variable, and E12, E13, E23 and E123 represent the crossed effects among variables. After simplification of the non significant factors, the particle size was adjusted to the response surface plotted in Fig. 8 and whose equation is given as follows (Eq. (12)):

Y1 ¼ 96:2 þ 1:32X1 þ 14:8X2  3:1X3  4:1X1X2  3:2X1X3

ð12Þ

This equation fits well with the experimental data as shown by the statistical test depicted in Table 9. Table 9 gives R2 value close to 1 and Fexp value larger than Fm1, m2 (=18.51) for all factors minus the interaction between the costabilizer and the surfactant concentrations (X2X3). Hence, based on F test and R2, Eq. (12) is considered to fit well with the experimental data. However, the range of validity of this model is claimed from 0.031 to 0.154 mol Laq1 for AIBN, from 0.056 to 0.308 mol Laq1 for hexadecane and from 0.015 to 0.031 mol Laq1 for SDS. Outside these bounds, experiments could not be fitted as well (Table 10). The mathematical model for the sonication time and power (X4 and X5) can also be established from the second DOE. Fig. 9 illustrates the corresponding surface response plot. The statistical analysis produces a R2 = 0.999 which means that it fits well with the experimental data.

Table 10 Extension of the DOE to other levels for the factors. Experiments

Fig. 8. Three-dimensional model for predicting particle size from the experimental conditions at AIBN concentration level +1.

KG04-16

Levels of experiments

Particle size (nm)

X1

X2

X3

Dz calculated

Dz measured

1.25

1

1

74

93

Table 9 Analysis of variance for the particle size*. Source Factor Factor Factor Factor Factor Factor Error Total

X1 X2 X3 X1X2 X1X3 X2X3

Sum of square

Degrees of freedom

SX1 SX2 SX3 SX1X2 SX1X3 SX2X3

13.91 1753.80 75.95 132.44 82.24 0.05

mX1 mX2 mX3 mX1X2 mX1X3 mX2X3

SR ST

0.07 2058.41

nR = N  1  N1

P P VXi VXiXj

Variance 1 1 1 1 1 1

VX1 VX2 VX3 VX1X2 VX1X3 VX2X3

2 7

VR = SR/nR

Fexp 13.91 1753.80 75.95 132.44 82.24 0.05 0.03

VX1/VR VX2/VR VX3/VR VX1X2/VR VX1X3/VR VX2X3/VR

Ftheo 408.45 51487.72 2229.81 3888.08 2414.39 1.55

m1 = mX1 ; m2 = nR m1 = mX2; m2 = nR m1 = mX2; m2 = nR m1 = mX1X2; m2 = nR m1 = mX1X3; m2 = nR m1 = mX2X3; m2 = nR SR/ST R2

18.51 18.51 18.51 18.51 18.51 18.51 0.003 0.997

* F and R2 are defined as F = VXi/VR and R = 1  SR/ST, where VXi is the mean square of regression obtained by dividing the sum of square (SXi) with the respective degree of freedom. VR represents the mean square error from the analysis of variance. SR and ST are the sum of square of regression and the total sum of square, respectively. R2 is the multiple correlation coefficient. A value close to 1 for R2 signifies a perfect fit to the experimental data.

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References

Fig. 9. Three-dimensional model for predicting particle size from the experimental conditions with the second DOE.

4. Conclusion Miniemulsion polymerizations of MMA were performed to study the effects of variables, including initiator, costabilizer and surfactant concentrations and sonication time and amplitude, on the particle size of the polymer. Full factorial design of experiments was effectively used to study the effects of interactions between these factors. The statistically based experiments plan enables an easier and closer interpretation of the results and more particularly an understanding of the influence of the links between the variables on both responses. The effects analysis indicates that the best approach for size control is varying the concentration of hexadecane and the surfactant while maintaining the initiator concentration steady. However, their interaction (X2X3) is not influent. The response surface for the sonication parameters implies the mutual dependence between the time and the amplitude of the sonication. To limit long time of sonication one should choose high sonication amplitude and fixed the time for the desired particle size. Mathematical models show that the prediction of particle size is achievable by using the statistic method of factorial design to quantitatively correlate the particle size with the synthesis conditions. Certainly, such a model is essential for producing particles with a targeted size on industrial scales. Acknowledgments We thank the Délégation Générale pour l’Armement (DGA) for financial support. Dr V. Madigou (University of Toulon) is gratefully acknowledged for TEM imaging.

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