I n d . Eng. Chem. Res. 1989,28, 65-74
65
Interparticle Monomer Transport in Miniemulsions Valmore
S.Rodriguez, J o a q u i n Delgado,? Cesar A. Silebi, and Mohamed S. El-Aasser*
Emulsion Polymers Institute and Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Theoretical predictions and experimental results are presented on the monomer transport between miniemulsion droplets of monomer type A and miniemulsion droplets of monomer type B. Styrene (St) and methyl methacrylate (MMA) are used as model monomers. The miniemulsions are prepared using sodium lauryl sulfate as surfactant and hexadecane as cosurfactant. The effects of cosurfactant concentration used in the preparation of the miniemulsions and the transfer area are analyzed. A theoretical model based on mass balances and equilibrium thermodynamics was developed to predict the monomer transport. The experimental monomer transport profiles agreed with the predictions of the mathematical model. However, the high resistance of the membranes used did not allow the determination of the mass-transfer coefficients. Emulsions can be categorized into three groups based on the size of the droplets, as well as the physical properties and characteristics. Conventional emulsions are referred to as “macroemulsions”and are usually prepared by mixing two immiscible liquids with one or more surfactants, ionic, nonionic, or a mixture of both. The dispersed liquid is in the form of droplets with diameter greater than 1.0 pm. Macroemulsions are opaque and milky in appearance and tend to separate on standing (Prince, 1977). “Microemulsions” are prepared using a combination of surfactants. Typically, a mixture of an ionic emulsifier and an alcohol (short-chain alcohol, usually, 1-pentanol or 1-hexanol) is used in their preparation. Microemulsions are optically isotropic, transparent, or translucent and consist of thermodynamically stable oil-in-water or water-in-oil systems with droplet sizes in the range 0.01-0.1 pm (Prince, 1977; Friberg, 1977; Shah et al., 1977). Mixed emulsifier systems, similar to those used in microemulsions, except that the emulsifier concentration is lower, have been used to prepare stable o/w (oil/water) emulsions with droplet sizes between 0.05 and 0.5 pm (Ugelstad et al., 1973; Delgado, 1986; Choi 1986). These emulsions are fluid, milky, and opaque in appearance, have droplet sizes between those of macroemulsions and microemulsions, and have been termed “miniemulsions”. The preparation of miniemulsions involves the use of a mixture of an ionic emulsifier and a long-chain alcohol or alkane as a cosurfactant and utilizes an emulsification technique developed at the Emulsion Polymer Institute of Lehigh University (Ugelstad et al., 1973, El-Aasser, 1985, Delgado et al., 1986). Miniemulsion copolymerization is a process in which particle generation and polymerization take place predominantly in stable comonomer emulsion droplets. The presence of a cosurfactant makes the miniemulsion copolymerization different from the conventional emulsion copolymerization in many respects, viz., droplet size, rate of polymerization, polymerization loci, and monomer swellability of the polymer particles (Ugelstad et al., 1973; Delgado et al., 1987, 1988a,b; Chamberlain et al., 1982). Particle morphology in emulsion copolymerization is known to be determined by the comonomer composition and concentration at the site of polymerization, as well as by the reactivity ratio and the water solubilities of the monomers. Obviously, the distribution of the monomers between the various phases (i.e., aqueous phase, monomer
* To whom all correspondence should be addressed. ‘Current address: 3M Center, Bldg 236-GB-16, St. Paul, MN 55144. 0888-5885/89/2628-0065$01.50/0
droplets, and monomer-swollen particles) throughout the course of the polymerization process is key to the particle morphology. Thus, knowledge of the transport mechanism and the rate of monomer transport between the various phases is essential for any study in emulsion copolymerization which is aimed at controlling particle morphology. Contrary to what was previously thought (Ugelstad et al., 1973), it seems that in a miniemulsion polymerization process not all the miniemulsion droplets initially present ultimately become polymer particles, and monomer transport through the aqueous phase was subsequently considered, since the monomer contained in the noninitiated monomer droplets must diffuse to the polymerizing particles (Delgado et al., 1988a,b; Chamberlain et al., 1982; Choi et al., 1985). Diffusional and interfacial phenomena in a conventional emulsion polymerization have been analyzed by Brooks (1970, 1971). Four possible loci of resistance to monomer transport were considered in this study: trwsfer from the droplets to the aqueous phase, transfer through the aqueous phase, transfer within the monomer-polymer particles, and transfer across the water-particle boundary. Brooks concluded that the diffusional processes which occur in the aqueous phase will not affect the course of polymerization and the only possible resistance to mass transfer, and consequently the only possible rate-determining step was the resistance to diffusion at the interface between monomer-swollen polymer particles and the aqueous phase. In the case of miniemulsion polymerization, the surfactant-cosurfactant combination could impose a significant resistance to the transfer of monomer from monomer droplets to the aqueous phase, due to a postulated interfacial complex with viscoelastic properties (Lack et al., 1985) or due to the higher level of emulsifier adsorption on the monomer droplets in the presence of the cosurfactant. Therefore, the resistance to mass transfer from the monomer droplets to the aqueous phase cannot be neglected without further information. Delgado et al. (1988a,b)developed a mathematical model to account for the monomer transport during the course of a miniemulsion copolymerization process. The model was applied to the miniemulsion copolymerization of vinyl acetate-butyl acrylate mixtures. The effects of the different components and parameters on the polymerization rate, monomer distribution, and copolymer composition were analyzed. It was concluded that the rate of polymerization gradually became monomer transport controlled if the value of the mass-transfer coefficient was lower than cm s?, resulting in a decrease in the polymerization 0 1989 American Chemical Society
66 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 Table I. Miniemulsion Recipe water monomer SLS hexadecane
180 g 60 g 10 mmol dm-3a 0-60 mmol dm-3a
OBased on the aqueous phase.
rate. The composition of the copolymer formed during the copolymerization was affected for values of mass-transfer coefficients lower than 5 X cm/s. The above study demonstrated the necessity of measuring the mass-transfer coefficients and to study further the rate and the mechanism of interparticle monomer transport. El-Aasser et al. (1988) studied copolymerizations starting from nonequilibrium conditions, where miniemulsions of individual monomers, butyl acrylate, styrene, and 2ethyl-hexyl acrylate, were mixed together and then polymerized after different mixing periods. The results indicated differences in the appearance and properties of the films formed from these latexes. These results assert the importance of the monomer transport in the miniemulsion copolymerization. Knowledge of the mass-transfer coefficients of the monomers across interfaces, mimicing those in existence under polymerization conditions, would allow prediction of the rate of monomer transport relative to the rate of copolymerization. Figure 1. Schematic diagram of' t h e compartmented diffusion cell.
Experimental Sect ion Materials. Styrene (St) and methyl methacrylate (MMA) monomers (certified grade Fisher) were washed with an equal volume of 10% sodium hydroxide solution. A separatory funnel was used to drain off the heavier aqueous phase and the procedure repeated until the aqueous phase remained clear. The monomers were washed several times with distilled and deionized water to remove the remaining base, until the wash water was neutral. The monomers were dried at 5 "C over anhydrous sodium sulfate (Fisher). The styrene was vacuum distilled under dry nitrogen a t approximately 20 Torr and 40 "C, the methyl methacrylate a t 60 Torr and 35 "C. The purified monomers were stored a t -2 "C until used. Sodium lauryl sulfate (SLS), 98% pure (Stepan Chemical Company), was purified by recrystallization from boiling ethanol, followed by extraction with anhydrous ethyl ether (certified grade Fisher), for 48 h, dried in a vacuum oven a t 40 "C for 5 h, and then stored under vacuum a t room temperature until used. Hexadecane (HD) (certified grade Fisher) was used as received. The water was distilled and deionized. Preparation and Characterization of the Miniemulsions. The miniemulsions of styrene and methyl methacrylate were prepared according to the following procedure using the recipe of Table I. When hexadecane was used as the cosurfactant, the hexadecane-monomer mixture was added to the aqueous solution of SLS and then subjected to agitation for 10 min followed by ultrasonification (Heat Systems Ultrasonic Cell Disrupter Model W-350) for 60 a t power 7 and 50% duty cycle a t room temperature. Stability Study. The shelf-life and centrifugational stability of styrene and methyl methacrylate miniemulsions were characterized as a function of hexadecane concentration. A sample of each miniemulsion was stored in a small bottle and its optical appearance monitored over a long period of time. Another sample of each miniemulsion was centrifuged a t 27000 rpm for a period of 1h in order to determine the amount of oil which separates and the amount of SLS remaining in the aqueous phase. The amount of SLS was determined by titration with Hyamine
1622 using the two-phase titration method with chloroform as the mixed indicator (disulphine blue/dimidium bromide) as described by Rosen and Goldsmith (1972). The stability studies were necessary to find the concentration of cosurfactant which imparted stability to the miniemulsions for a period of time a t least as long as the time required to run the experiments. Miniemulsion Droplet Size. Miniemulsion droplet size was determined by using a Coulter N4 submicron particle analyzer. This instrument requires dilution of the miniemulsion, so the serum needed for the dilution was prepared by separating the clear aqueous phase formed by centrifuging another portion of the same miniemulsion. This procedure was used in order to prevent diffusion of the monomer from the miniemulsion droplets to the aqueous phase during dilution. Studies of Monomer Transport in a Compartmented Diffusion Cell. The compartmented diffusion cell shown in Figure 1 consists of two chambers with a capacity of 200 mL each, separated by a semipermeable membrane. Each compartment has a sampling port from which samples are withdrawn during the experiment. The membrane is placed in the cell in a such manner that both sides of the membrane are completely covered by the miniemulsions, even after samples have been withdrawn from the cell. Two miniemulsions are separated by a semipermeable membrane which allows the diffusion of monomer between the two compartments but not the diffusion monomer droplets or monomer-swollen particles. During the monomer transport experiment, the diffusion cell is rotated along its axis at a speed of 175 rpm. As monomer transport takes place, each chamber is sampled and analyzed by gas chromatography (5890A Hewlett-Packard with 3393A Hewlett-Packard Integrator). The amount of sample withdrawn each time is 0.25 mL from each chamber, so that the volume changes in the cell can be considered negligible. Nevertheless, the volume changes were considered in the analysis of the monomer transport. The monomer transport rate and the time needed to reach equilibrium are determined.
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 67
STYRENE MMA
VI
3
0
w
3
a 4
Figure 2. Appearance of methyl methacrylate miniemulsion after 3 months, as a function of the hexadecane/SLS ratio noted on the sample bottles (1-6), based on 10 mmol dm-3 SLS.
u
8 v)
4 VI
d W co
60.0
GI H
0
2.0
4.0
6.0
8.0
HEXADECANE/SLS (MOLAR RATIO)
tT
Figure 4. Concentration of sodium lauryl sulfate in the aqueous phase as a function of hexadecane/SLS ratio used in their preparation, based on 10 mmol dm-3 SLS; see Table I.
F
1
40.0
I
O\O
0. 0.
100.0
80.0
2.0
u
Table 11. Effect of Hexadecane Concentration on the Miniemulsion Droplet Size, Determined Using the Coulter N4 Particle Size Analyzer run" droplet size, nm std. dev, 7% MMABO 124 50 MMA40 101 65 MMAGO 91 32 72 Sty20 226 sty40 175 38
" The number represents the concentration of hexadecane used in the miniemulsions. 0.
I 0.
I
I
2.0
I
I
4.0
I
I
6.0
I
I
I
8.0
HEXADECANE/SLS (MOLAR RATIO) Figure 3. Amount of methyl methacrylate separated (oil separated/initial) as a function of hexadecane/SLS ratio, based on 10 mmol dm-3.
Results and Discussion Stability Studies. The appearance of the methyl methacrylate miniemulsions was examined 3 months after preparation. Figure 2 shows that the creamed layer in the miniemulsion decreased as the HD/ SLS ratio increased, which indicates as increase in the emulsion stability against creaming. Upon centrifugation of the miniemulsion, the amount of MMA separated also decreased with increasing HD/SLS ratio, as illustrated by Figure 3. Figure 4 shows the concentration of SLS remaining in the aqueous phase after centrifugation of methyl methacrylate and styrene miniemulsions as a function of HD/SLS ratio. For both miniemulsions, as the HD/SLS ratio increased, the concentration of SLS in the aqueous phase decreased; i.e., the amount of SLS adsorbed onto the droplets increased. These results suggest that increasing the HD/SLS ratio results in a decrease in the droplet size of the miniemulsion and consequently their stability against creaming and centrifugation is increased. Also it is shown in Figures 3 and 4 that a plateau is reached for the amount of oil separated upon centrifugation and the concentration of SLS in the aqueous phase at a HD/SLS ratio of 4.
Table I1 shows the variation of the miniemulsion droplet size with the concentration of hexadecane in the miniemulsions, using the Coulter N4 particle analyzer. The results obtained with the Coulter N4 are in good qualitative agreement with the results described in the emulsifier adsorption analysis. As suggested previously, the miniemulsion droplet size decreases when the concentration of hexadecane in the miniemulsion is increased. It is also noted that the miniemulsions have a broad distribution, as has been found by other investigators (Ugelstad et al., 1973; Delgado, 1986; Choi, 1986). Mat hematical Model. The mathematical model developed by Delgado et al. (1988a,b) was modified, so as to be applied to the two-chamber reactor system, each chamber containing one type of monomer miniemulsion separated from the other by a semipermeable membrane. It was postulated that, in an agitated system in the absence of a chemical reaction, the mass of compound i transported per unit time from one phase to the adjacent phase is proportional to the transfer area, A, and the difference between the equilibrium concentration of compound i in the phase under consideration (concentration that would be at equilibrium with the concentration of compound i in the adjacent phase), Cei, and its actual concentration (bulk concentration), Ci. The transport equations for the first monomer can be expressed as follows: monomer transport from the droplets in chamber 1 to the aqueous phase in chamber 1
68 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989
monomer transport from the aqueous phase in chamber 1 to the aqueous phase in chamber 2 through the membrane wi,al-a~/dt
= KmDia(Ci,al -
Ci,a2)
(2)
monomer transport from the aqueous phase in chamber 2 to the droplets in chamber 2
- Ci,d2)
(3) where dNi,d/dt is the molar flow of monomer i (mol s-'), Ki,dl-?l and Ki,a2+ Icm s-l) are the overall mass-transfer coefficients of monomer i between the droplets and the aqueous phase in chamber 1and chamber 2, respectively, Diais the diffusion coefficient (cm2s-l ) of monomer i in water, a refers to aqueous phase, a1 and a2 refer to the aqueous phases in chambers 1and 2, respectively, d l and d2 to the droplet phases in chambers 1and 2, respectively, and K, is the membrane constant. The membrane constant is defined as (Geankoplis, 1983) wi,dZ/dt
= Ki,d2-a2Ad2(Cei,d2
algorithm, the initial concentration of the components in each phase and each chamber are calculated by applying the thermodynamic equilibrium condition to each chamber separately. The equilibrium concentrations for the whole system during the transport process are calculated by applying the thermodynamic equilibrium condition to the whole system (two chambers). Having determined values of the equilibrium concentration, the differential equations expressing the monomer transport process are solved by conventional numerical techniques. The partial molar free energy of mixing of component i in a given phase p is given by the Flory-Huggins lattice theory (Flory, 1953), with the addition of an interfacial energy term when the monomer phases are in the form of spherical droplets (Morton et al., 1954), as expressed by Ugelstad et al. (1980):
(4)
where A , is the membrane area, c the fraction of area open,
K, the tortuosity factor, and 6 the thickness of the membrane. Equivalent expressions (eq 1-3) can be written for the second monomer present in the system. The equations described above are used in conjunction with the mass balance and the volume change for each phase. mass balance equation
+ Ni,al + Ni,aP + Ni,d2 = N i volume in each phase Ni,dl
(5)
i=l n
(7)
n Va2
=
C ViNi,a2
i=l
(9)
where V is the volume of the phase under consideration and Vi is the molar volume of component i. The equilibrium concentrations of the monomers in the various phases needed to solve the differential equations can be determined from equilibrium swelling thermodynamics (Ugelstad et al., 1980). Once the monomer transport process begins, the equilibrium condition, where the net flow of the monomers is zero, is attained when the partial molar free energies of mixing of each monomer in the different phases are the same. This condition is expressed by a set of nonlinear equations for the thermodynamic equilibrium and the mass balance for each component, (10)
where xl, is the Flory-Huggins interaction parameter; @l,q is the volume fraction of component i in phase q; m, is the ratio of the equivalent number of molecular segments between i and j , usually expressed by the ratio of the molar volumes ( V r /VI);R is the gas constant; T is the temperature; r is the radius of the phase under consideration; and y is the interfacial tension. The use of the Flory-Huggins lattice theory to express the partial molar free energy of mixing in mixtures involving small molecules, as in the case of the monomer droplets and the aqueous phase, is open to doubt. Ugelstad et al. (1980) proposed that the use of the Flory-Huggins equation was valid provided the values for the interaction parameter (x,) and the ratio of the equivalent number of molecular segments (m,) are those obtained experimentally (x*~],m*ll). It should be pointed out that under these circumstances m*, is not necessarily equal to the ratio of the molar volume of the two components, and consequently, they have to be obtained experimentally. With this in mind, the expressions accounting for the partial molar free energy in every phase are as follows (from eq 12): (i) monomer droplets
x*12@2,d2
+ X*1H@H,d2 + @2,d@H,d(x*l2 +
- X*z~m*12) + 2YdVi/(rdRT)
(14) where d stands for monomer droplets 1 and 2 for monomers 1 and 2, H for hexadecane, and Y d is the interfacial tension between the monomer droplets and the aqueous phase. Equation 14 is used for both styrene and methyl methacrylate monomers. (ii) continuous aqueous phase x*lH
(E)
+ (1- m * 1 2 ) 4 2 , a + (1- m * l w ) @ w , a + ~ * 1 2 @ 2 , a 2+ X*lw4w,aP + + 2 , a @ w , a ( ~ * 1 2 + x * l w - ~ * 2 w m * 1 2 ) l,a
where ( G / ( R T ) ) i , p is the partial molar free energy of mixing of component i in phase p , and is calculated from equilibrium swelling thermodynamics. In the integration
= In
(15) where a stands for the continuous aqueous phase and w for water. Equation 15 is for monomer 1 (styrene), and a similar expression is used for monomer 2 (MMA). All the interaction parameters, x*i,'s, and the ratio of the equivalent number of molecular segments, m*ij, be-
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 69 Table 111. Partitioning of MMA and ST in Water 61,d
&,a
SS?
1.0000 0.7644 0.5420 0.5131 0.2646
0.00040 0.000 32 0.000 21 0.000 20 0.000 07
0.0359 0.0290 0.0189 0.0178 0.0067
and
S M W
1 0 4 ~ ~ 4.00 4.18 3.87 3.89 2.64
62.d
62,a
0.2356 0.4579 0.4868 0.7353 1.0000
0.003 56 0.007 56 0.008 10 0.011 34 0.017 60
M sM A a
1 0 4 ~ ~
0.3369 0.7189 0.7698 1.0813 1.69402
151.1 165.0 166.4 153.2 176.0
in g/100 g of HZO.
tween i and j were obtained by using the procedure described by Ugelstad (1983) and Delgado (1986). Partitioning of MMA and St in the Presence of Each Other. The concentration of MMA and St in water were determined as a function of the MMA-St ratio in the organic phase. Three grams of organic phase with different MMA-St ratios was dispersed in 9 g of water in the absence of surfactant and allowed to equilibrate. The aqueous and the organic phases were separated by centrifugation (Servall, RC-5B refrigerated superspeed centrifuge, Du Pont Instrument). The concentrations of the monomers In water were determined by gas chromatography using a Hewlett-Packard 5890 gas chromatograph equipped with HP-17 column, 10 m long and 530 pm in diameter. The results given in Table I11 shows that, while the water solubility of St (SSJ decreases as its equilibrium volume fraction in the organic phase (@l,d)is decreased, the water solubility of MMA ( S M W ) increases. On the other hand, the partition coefficient for St ( K J ,given by the ratio of the equilibrium volume fractions of the monomer in the aqueous phase and in the organic phase, follows the trends of the solubilities of the monomers in the aqueous phase. As the solubility of the monomer in the aqueous phase increases, its partition coefficient also increases. This is obeyed by both St and MMA monomers. The behavior of the solubilities and the partition coefficient is a consequence of the different interaction between monomers and between monomer and water. These data have been used to calculate the monomer-water and monomer-monomer interaction parameter using the following treatment. If two partially miscible phases, for example, monomer and water, are placed in contact, components in the system will be present in both phases due to the partitioning established by the thermodynamicequilibrium condition. In the following analysis, water is not considered to be present in the organic phase to be consistent with the model. Under this condition, when the two phases are at equilibrium
(3) =(")RT RT i,d
i,a
where i = 1 for St and i = 2 for MMA. The relationship resulting from eq 16 can be transformed into the following expression after rearrangement
Table IV. Partitioning of St in the Presence of Hexadecane 6l.d 61.. Sst 1 0 4 ~ ~ 0.000 40 0.000 28 0.000 22 0.000 16 0.000 11
1.0000 0.7094 0.4516 0.2179 0.1138
0.0359 0.0249 0.0207 0.0152 0.0104
4.00 3.94 4.86 7.43 9.71
Table V. Partitioning of MMA in the Presence of Hexadecane 1.0000 0.7881 0.6283 0.4369 0.2193
0.0176 0.0126 0.0124 0.0111 0.0087
1.694 1.200 1.180 1.060 0.830
176 159 197 254 378
The values x*lwand x*%needed to calculate Ai, Bi,and Ci may be estimated from the water solubility of the monomer in the absence of the comonomer and using the corresponding thermodynamic expression. These values are x*lw= 7.94 and x*%= 4.05 with values for m*lw = 1.11 and m*2w= 0.87. These parameters are very similar to those determined by Ugelstad (1983). Plotting B2/A2versus C 2 / A 2will give a straight line with slope m*21and intercept x*21.The values obtained are x*21 = 0.31 and m*21= 1.01 where x*12= ~ * ~ ~ and / mm*12 * ~ ~ = l/m*21. Partitioning of St and MMA in the Presence of Hexadecane. Hexadecane is present in the miniemulsion droplets, and thus the thermodynamic parameters, m*ij and x*ij, between monomer and hexadecane are also needed. Mixtures of each monomer and hexadecane were dispersed in known amounts of water, and the concentration of the monomer in the aqueous phase after equilibration was determined by gas chromatography as described previously. The results in Tables IV and V show that the solubility of both monomers in water is decreased upon the addition of hexadecane to the organic phase. On the other hand, for both St and MMA, the partition coefficient between aqueous and organic phases increased with increasing volume fractions of hexadecane. This effect is probably the result of the very low water solubility of hexadecane, which contributes to the free energy of mixing of the monomers in the aqueous phase to a small extent. Similar results were obtained by Delgado et al. (1988a,b) using the same analysis for VAc and BuA. The thermodynamic parameters, m*ij and x*ij, for monomer and hexadecane were determined by using a similar approach used to obtain monomer-monomer interaction parameters. When hexadecane is present in the system, the expressions for the free energy of mixing of monomer i in the oil phase and aqueous phase can be written as
(g)i,d
= In
@i,d
+ (1 - m*iH)+H,d + x*iH@H,d2
(21)
70 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 Table VI. Values of the Interaction Parameters, Segment Volume Ratios, Interfacial Tensions of the Oil Compound with Water, and the Resistance of the Membrane Used in the Simulation x*1.2 x*2,1 XI,€{
X2,H X*l,W
x*2,w
71
x*2,1/m*1,2
m*1.2
0.31 1.69 2.80 7.94 4.05 3.41 4.86 169
m*2.1
ml,H m2,H
m* 1 ,W rn*2,w 72
(KmD1J1
-I
L
1
N
a 0.24 U
0.16 3
IL
LL
0.08
0
2 r
,HAi
Ai
i
1
H
0.
Applying the thermodynamic equilibrium condition and rearranging yields - -Ci = x * +~m*. ~ Bi -
7
W
m I a I
l/m*2,1 1.01 1.54 1.83 1.11 0.87 3.31 253
r----T--
0
(23)
where
L-
- L A . L L - L A - - . J
0.
40.0
80.0
120.0
160.0
200.0
TIME (MINI
Figure 5. Amount of methyl methacrylate diffused to chamber 2 as a function of time, varying the ratio of hexadecane (chamber l/chamber 2) in the range 1-6 (as indicated on curves). The hexadecane concentration in chamber 2 was kept constant a t 20 mmol dm+.
Ai = d'H,d2 0.40
R, = -4 H ,d 6,d
cc = In - + $H,d 4i,n
-- &,a
+ m*twd'w,a
- X*zw&,a
(26)
- C , / A , plotted against B J A , resulted in a straight line with slope m*IHand intercept x*,H. The values obtained are m*,H= 1.538, x*lH = 1.694, m*2H= 1.826, and x*2H= 2.801. To account for the change in the interfacial tension of the droplets with time during the monomer transport, the following expression was used: " (27) Y cri4, i=l
where yi (dyn cm-l) is the interfacial tension between any one of the oil compounds in the system and water, obtained by the drop volume method (Tseng, 1983). Table VI shows the values of the different parameters used in the simulation. Simulation Results of Monomer Transport in the Compartmented Diffusion Cell. Effect of Hexadecane Concentration on Monomer Transport. The mathematical model described previously was applied to simulate the monomer transport between St and MMA droplets as a function of the amount of cosurfactant (hexadecane) used to prepare each miniemulsion. The following assumptions were made in order to simplify the calculations. 1. The membrane offers zero resistance (K, = a),which implies that at any time the concentration of monomer i in the aqueous phase in chamber 1 is the same as the concentration of monomer i in the aqueous phase in chamber 2. This represents the case when two miniemulsions are mixed without any membrane separating them. 2. The miniemulsions are assumed to be monodisperse in droplet size, with a diameter of 180 nm. This value was chosen based on the values obtained previously by several investigators (Ugelstad et al., 1973, Delgado, 1986; Choi, 1986) and the values obtained in the present work. 3. The mass-transfer coefficients for both monomers were assumed to be the same and equal to IO4 cm sd. This value is arbitrary. 4. The concentration of HD in the aqueous phase was considered to be zero, and thus the amount of HD in each compartment remained constant. Figures 5 and 6 show the model predictions for the amount of MMA diffusing, as a function of time, from
i-L'j 0.08
I -
0.1
0.
'
'
40.0
'
'
80.3
'
'
'
I
'120.0 160.0 200,J
TIi4E (MIi'i)
Figure 6. Amount of styrene diffused to chamber 1 as a function of time, varying the ratio of hexadecane (chamber l/chamber 2) in the range 1-6 (indicated on the curves). The hexadecane concentration in chamber 2 was kept constant a t 20 mmol dm-3.
chamber 1 (initially containing a MMA miniemulsion) to chamber 2 (initially containing a styrene miniemulsion) and the amount of styrene diffusing to chamber 1 from chamber 2, respectively. The amount of hexadecane in the MMA miniemulsion was varied in the range 20-60 mmol dm..3, while the amount of hexadecane in the styrene miniemulsion was kept constant a t 20 mmol dm-3. The monomers concentration profiles in Figures 5 and 6 demonstrate the two roles played by hexadecane in these systems: first as a retarder for the diffusion of monomer out of the droplets in which it is residing due to its lack of water solubility (Higuchi and Misra, 1962), and secondly as a swelling promoter due to thermodynamic effects, mainly entropic in nature (Ugelstad et al., 1980). As the amount of hexadecane in the MMA miniemulsion is increased, the amount of MMA at equilibrium in the styrene miniemulsion droplets is diminished, reducing the overall mass-transfer rate. On the other hand, the amount of styrene diffusing from chamber 2 to chamber 1 increases as the amount of hexadecane in the MMA miniemulsion droplets in chamber 1 increases. This increasing amoml. of styrene in the MMA droplets has an interesting effect on the shapes of the curves in Figures 5 and 6. Figure 5 shows that the amount of MMA in chamber 2 containing the styrene miniemulsion increases up to a maximum and
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 71 Table VII. Effect of the Initial Droplet Size of the MMA Miniemulsion, Originally in Chamber 1, on the Amount of Monomer and Final Droplet Size at Equilibrium in Chambers 1 and 2 chamber 1 2 _ _ _ chamber ~ -initial final initial final droplet size, amt a t equilib droplet size, droplet size, amt a t equilib droplet size, nm MMA, mol St, mol nm nm MMA, mol St, mol nm 180 160 140 114
0.397 0.393 0.387 0.375
0.400 0.396 0.390 0.375
216 192 166 132
then decreases with time. Increasing the hexadecane content in the MMA miniemulsion in the range 20-60 mmol dm-3 decreases the amount of MMA at the maxima and increases the diffusion rate of MMA back to chamber 1 (beyond the time required to reach the maxima). Once the mass-transfer process occurs, the equilibrium state will be given by an equality in the composition of the droplets in terms of concentrations. Upon increasing the HD content in the MMA miniemulsion in chamber 1 due to the insolubility of the hexadecane in water and therefore its imposed immobility, more styrene will have to be transported from chamber 2 to chamber 1than MMA from chamber 1 to chamber 2, to have equal compositions in both types of droplets at the final state. Consequently, the profile of the number of moles of the monomers transported between the chambers shows a maximum in the amount of MMA being transported to chamber 2, as a result of the different concentration gradients between monomers. By the time an amount of MMA equivalent to what should be present in chamber 2 in the final equilibrium state has been transported, the concentration of MMA in the droplets of chamber 2 (initially containing styrene miniemulsion) is actually less than that of the equilibrium state a t that point in time. This is because not all of the styrene required to reach equilibrium in chamber 1 has yet been transported. The reason for this is mainly due to the difference in hexadecane concentration in the two miniemulsions. As a result, MMA continues to diffuse up to a maximum. Thereafter, it is transported back to chamber 1,because at this point there is a negative gradient in the MMA concentration. This arises from the continuous transfer of styrene from chamber 1to chamber 2. This particular type of diffusion may be seen experimentally, as will be shown later, despite the high resistance of the membrane, because this behavior is mainly a consequence of the equilibrium concentrations and not of the mass-transfer coefficients. Effect of the Droplet Size on the Monomer Transport and Equilibrium Amount. Simulations showing the effect of the initial droplet size of the miniemulsions on the monomer transport are shown in Figures 7 and 8. The concentration of hexadecane in the MMA miniemulsion was set to be 40 mmol dm-3, and in the styrene miniemulsion 20 mmol dm-3. The droplet diameter was varied in the range 114-180 nm for the MMA miniemulsion but remained constant a t 180 nm for the styrene droplets. The final equilibrium concentrations are not shown in Figures 7 and 8 due to the relatively long time required to reach equilibrium. Table VI1 shows the results of the equilibrium amount of both monomers in chambers 1 and 2 and the corresponding final droplets sizes at equilibrium. The simulation results show that when the droplet size of the MMA miniemulsion (originally located in chamber 1) increases, the total amount of monomer in chamber 1 increases, thus decreasing the amount of monomers in chamber 2. Effect of Mass-Transfer Coefficients of the Monomers on the Monomer Transport. The third parameter
180 180 180 180
0.073 0.077 0.083 0.095
0.073 0.077 0.083 0.096
121 122 126 132
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Figure 7. Amount of methyl methacrylate diffusing to chamber 2, containing styrene miniemulsion, a8 a function of time varying the droplet diameter of methyl methacrylate miniemulsion in chamber 1. The concentration of HD in the styrene miniemulsion was 20 mmol and 40 mmol dm-3 in the MMA miniemulsion. I
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Figure 8. Amount of styrene diffusing to chamber 1, containing methyl methacrylate miniemulsion, as a function of time, varying the droplet diameter of miniemulsion in chamber 1. The concentration of HD in the styrene miniemulsion was 20 mmol dm-3, and 40 mmol dm-3 in the MMA miniemulsion.
investigated with the mathematical model was the effect of the ratio of the overall mass-transfer coefficients of the two monomers. For this study, the mass-transfer coefficient for styrene (Kst) was kept constant at an arbitrary value of cm s-l, and the mass-transfer coefficient for MMA (KMVIMA) was varied from 1.0 to 2.0 times the masstransfer coefficient of styrene. The droplet size distributions were again considered to be monodisperse with a diameter of 180 nm for both miniemulsions. The amount of hexadecane in both miniemulsions was 20 mmol dm-3. Figures 9 and 10 show the results of the simulations under these conditions. The rate of MMA transport from chamber 1 to chamber 2 increases with increasing KMM, as expected, due to the fact that the rate of monomer transport is directly proportional to the mass-transfer coefficient. As KMMA is increased, the initial rate of
72 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 I
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Table VIII. Membranes Tested in the Compartmented Diffusion Cell open pore size, St brand name sumlier area. 70 um diffusion polyester Nucleopore 6-10 0.1 no MSI
