Langmuir 1991, 7, 73-80
73
Swelling of Latex Particles by Water-Soluble Solvents. 2. Thermodynamic Equilibrium Analysis R. Popli Louis Laboratory, S. C. Johnson and Son, Inc., Racine, Wisconsin 53403 Received March 22, 1990. I n Final Form: June 28, 1990 The swelling of an acrylate terpolymer latex and a PS latex, by solvents with varying degree of solubility in water, was analyzed by using the BGDP thermodynamic equilibrium model. The agreement between experimentaldata and model predictions is satisfactorybut not precise. The sensitivityof model predictions to variations in the interaction parameters was examined. The water-solvent interaction parameter is a critical factor in determining the partition behavior for many solvents. work is to examine (a) the swelling data for water-soluble solvents, described in part 1, in context of the BGDP equilibrium model, and (b) the influence of various model parameters on the equilibrium swelling predictions for the water-soluble solvents. A brief description of various equilibrium models is given in the background section and is followed by a detailed analysis of the data.
Introduction The interest in controlling and modifying latex polymerization and thereby the latex properties has been a major thrust behind the interest in study of latex particle swelling by low molecular weight compounds (monomers or solvents).14 A secondary, nevertheless important, industrial application is a coalescing solvent addition to aid film formation in water-based coating^.^^^ The solvent partitions between the polymer and the aqueous phase of a latex. The partition equilibrium changes as solvent to water ratio in the svstem changes ., continuouslv during film drying.’ Morton.’ Kriabaum,2G a r d ~ nand , ~ U g e l ~ t a dcarried ~~~~l out some of the early’experimental work and developed models to describe the swelling of latex particles by solvents with low degree of solubility in water. The word solvent is used here to refer to a monomer or a solvent. The extension of these models to include the solubility of solvent in aqueous phase has been done in recent years by Guillot,12Tseng,lsJ4 Bindschaedler,’5-17etc. An excellent review of the literature in the area was done by Ugelstad.18 The latest model due to Bindschaedler et al., referred henceforth as the BGDP model, takes into consideration the influence of solvent concentration on water-solvent interactions. The BGDP model has also been shown to be a general case of other m0de1s.I~The purpose of this
Background Morton’s treatment is based on a balance of change in the surface energy and the free energy of interaction of the swollen polymer particle.’ This balance is represented by the equation
I
and
(1)Morton, M.; Kaizgermann, S.; Altier, M. W. J. Colloid Sci. 1954, 9,300. (2)Krigbaum, W.R.;Carpenter, D. K. J. Polym. Sci. 1954,14,241. (3)Gardon, J. L. J. Polym. Sci., Part, A-1 1968,6,2859. (4)Ugelstad, J.; El-Aasser, M. S.; Vanderhoff, J. W. J. Polym. Sci., Polym. Lett. 1973,11, 505. (5)Dillon, R.E.; Bradford, E. B.; Andrews, R. D., Jr. Ind. Eng. Chem. 1953,45,728. (6)Hansen, C. M. Prog. Org. Coat. 1982,10,331. (7)Hoy, K. L.J. Paint Technol. 1973,45,51. (8)Ugelstad, J.; Hansen, F. K.; Lange, S. Makromol. Chem. 1974,175, 507. (9)Ugelstad, J. Makromol. Chem. 1978,179, 815. (IO)Ugelstad, J.; Kaggerud, K. H.; Hansen, F. K.; Berge, A. Mukromol. Chem. 1979,180,737. (11)Ugelstad, J.; Kaggerud, K.; Fitch, R. M. Polym. Colloids 2, [Proc. Symp. Phys. Chem. Prop. Colloidal Part.], 1980,83. (12)Guillot, J. Makromol. Chem., Rapid Commun. 1980,1, 697. (13)Tseng, C.M.; El-Aasser, M. S.;Vanderhoff, J. W. Org. Coat. Plast.
R = (&/$J/(v’i/v’J (3) The subscripts 1, 2, and 3 correspond to solvent, water, and polymer, respectively. R is calculated from the expression -log ( R ) = (1- X,) log ($z/v’2)
--.
+ 2x12(42 - v’z) +
+ x13 - x1x23)43 (4) x’s are the interaction parameters, $3 is the polymer volume fraction, and X I is the ratio of the molar volumes of the solvent to water. This equation can be used to calculate the swelling of a polymer, if $3 is known and is nearly constant. Gardon has applied this equation to the swelling of latex particle^.^ The model of Tseng et al. takes into account water presence in the polymer phase.13J4 Ugelstad and Bind-
Chem. 1981.45. - - - - , 317. --
(XI2
(14)Tseng, C. M.; El-Aasser, M. S.; Vanderhoff, J. W. ACS Symp. Ser. 1982. No. - - 197. 197. (15)Bindschaedler, C.;Gurny, R.; Doekler, E.; Peppas, N. A. J. Colloid Interface Sci. 1985,108,75. (16)Bindschaedler, C.; Gunry, R.; Doekler, E.; Peppas, N. A. J. Colloid Interface Sci. 1985,108,83. (17)Bindschaedler, C. PhD Thesis, University of Geneva, 1984. (18)Ugelstad, J.; Mork, P. C.; Mfutakamba, H. R.; Soleimany, E.; Nordhuus, I. Science and Technology ofPolymerColloids;NATO AS1 Series; Poehlein, G. W., Ottewill, R. H., and Goodwin, J. E., Eds.: Plenum: New York, 1983;Vol. 51.
----.-
+ $3 = 1.0
(2) where $1 is the volume fraction of solvent within the latex particle, $3 is the volume fraction of polymer within the latex particle, xi3 is the solvent-polymer interaction parameter, V1 is the molar volume of the solvent, ro is the latex particle radius, y is the interfacial energy, R is the gas constant, and T is the temperature. This model shows good agreement with the partition data for strong hydrophobic solvents or monomers. Krigbaum and Carpenter’s model allows a consideration of the interaction between solvent and watera2 A ratio “R”,defined in terms of the volume fractions of two liquids, $1 and $2 in the polymer phase and v f l and v’2 in the binary liquid phase, is calculated $1
- - . I
@
1991 American Chemical Society
Popli
74 Langmuir, Vol. 7,No. 1, 1991
schaedler point out that the solvent-water activity in the aqueous phase makes a strong contribution to the partition behavior for water-soluble solvents, and Tseng's model does not clearly define how this is to be taken into account.15J8 It is also pointed out that Tseng et al. assume a constant value for the solvent and water interaction parameter, which is not true unless the aqueous phase is saturated by an excess of solvent. The BGDP model is realistic and takes into account various interactions of the system.'5-" Assuming no polymer from the polymer phase is dissolved in water, the solvent/water and the solvent/water/polymer are the two phases in equilibrium. The following equations are derived for the swelling of polymer latex particles: log 41 + 1- $1 -
43 x,- x, + (g1242 + x1343)(42 + 43) 42
Table I. Vapor-Pressure Data: Ethanol-Water and DEGEE-Water Systems ethanol wt headspace DEGEE wt headspace DEGEE vapor fraction in ethanol vapor fraction in solution
fractiona (23 "C)
solution
fractiona (80 "C)
0.022 0.049 0.100 0.200 0.300
0.068 0.105 0.115 0.210 0.281
0.020 0.050 0.100 0.200 0.300
0.006 0.014 0.028 0.031 0.046
0 Head space vapor fraction is the vapor pressure measured over the solution to that measured over the pure solvent.
However, it is included here for sake of clarity and to describe in detail the method used to evaluate the watersolvent interaction parameter. The latter method is different from the one used in refs 15 and 16. BGDP Model Parameters. Water-solvent Interaction Parameter, 812. BGDP used a procedure employed by Konigsveld20 and Kamide21 to examine the concentration dependence of solvent-polymer interaction parameter in polymer solutions. For the water-solvent mixture, the interaction parameter is assumed to be a linear function of water concentration (10) = XO(1 + P V ' J where xo a n d p are parameters defining the concentration dependence of ~ 1 2 .BGDP estimated xo and p from the solution compositions at the critical points in phase equilibria of solvent-water mixtures. This procedure gives the following equation for concentration dependence of x12 for the water-DEGBA solutions: XI2
1- ~
&lZ - d1X2+ X&12~'12+ X2~'12u'2dv' (6)
' 2
2
(7) 41 + 42 + 43 = 1 Again the subscripts 1,2, and 3 correspond to solvent, water, and polymer, respectively. u'1 and u'2 are the volume fractions of solvent and water in the aqueous phase. 41, 42, and 43 are the corresponding volume fractions in the polymer phase. xi3 and ~ 2 are 3 the solvent-polymer and the water-polymer chi interaction parameters, respectively. g12 is the water-solvent interaction parameter and is concentration dependent. ro is the radius of unswollen polymer latex particle, and VI is the molar volume of solvent. u1 and u2 in the above equations are defined as = &/(#I 42). Also, ~1 + UP = 1. If there is no water in the polymer phase, $2 = 0 and eqs 5 and 6 are simplified to
+
-v'2 + gl2U';
xz
- ut u'
- (8) dd2
Also
+ 43 = 1
(9) Following the terminology used in ref 15, eqs 5,6, and 7 describe the ternary, and eqs 8 and 9 the pseudobinary model. 41
Model Predictions, Data Analysis, and Discussion The BDGP model is used to analyze the experimental data presented in part 1of this series.19 A brief description of the computer program used to solve eqs 5,6, and 7 or eqs 8 and 9 is presented in the Appendix. The following description of BGDP model parameters is along the lines discussed in refs 15 and 16 and may seem repetitious. (19) Popli, R.; Luccas, M. H.; Tsaur, S. L. Langmuir,preceding paper in this issue.
xlZ= 52.1 - 3 9 . 1 ~ ' ~ However, this procedure cannot be applied to completely water soluble solvents. In these cases, a direct measurement of the headspace concentration of the solvent vapor in equilibrium with the solvent-water mixture is obtained at various solvent concentrations. Assuming ideal gas conditions, xl2 values are obtained from the solvent activity, "a", using Flory-Huggin's equationz3 p1- Po log a = - log d1
RT
+
U'l
+ X12(l - U'J2
(11)
where p1 and po are chemical potentials of the solvent in solution and of the pure solvent, respectively. XZis the ratio of molar volumes of water and the solvent, and u'1 is the volume fraction of solvent in the mixture. The equilibrium vapor pressure data for the ethanolwater system was obtained at 23 "C. These dataare shown in Table I and are in agreement with the literature values on this system.24 The measurements for solvent DEGEE were done at 80 "C due to the low solvent vapor pressure at room temperature. An estimate of x variation with temperature using the UNIFAC method25126 shows a relatively weak temperature dependen~e.~'The vapor (20) Konigsveld, R.; Steverman, A. J. J . Polym. Sci., Part A-2 1968, 6, 325. (21) Kamide, K.; Sugamiya, K. Makromol. Chem. 1970,139, 197. (22) Kamide, K.; Sugamiya, K.; Kawai, T.; Miyazaki, Y. Polym. J. 1980, 12, 67. (23) Flory, P. J. Principles of Polymer Chemistry; Cornel1 University Press: Ithaca, NY, 1953. (24) Hall, D. S.; Mash, C. J.; Pemberton, R.C. NPL Report CHEM 95: National Phvsical Laboratorv, Division of Chemical Standards: Teddington, England, January 1979. (25) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. AIChEJ. 197S, 21, 1086. (26) Gmehling J.; Rasmussen, P.; Fredenslund, A. Ind Eng. Chem. Process Des. Deu. 1982, 21, 118.
(27) Popli, R. Unpublished results.
Swelling of Latex Particles
Langmuir, Vol. 7, No. 1, 1991 75
3.5--
w
3
i
5
3.0-
s,
H
/
1
2.04 0 0
0.1
0.2
0.3
I
1.
0.4
0.00
0.
io
0.20
0.1
0.2
C 3
SOLVENT VOLUME F R A C T I O N
SOLVENT VOLUME F R A C T I O N
Figure 1. Chi parameter versus solvent volume fraction for ethanol-water system.
5.04
0
0.30
SOLVENT VOLUME F R A C T I O N
Figure 2. Chi parameter versus solvent volume fraction for DEGEE-water system. pressure data for the DEGEE-water system is also shown in Table I. The x value estimates for MEK-water were obtained from the published literature data obtained at 75 0C.28 Again the temperature dependence of x12 values was estimated to be The plots of x 1 2 dependence on solvent concentration for ethanol, DEGEE, and MEK are shown in Figures 1, 2, and 3,respectively. A linear fit of the data was obtained, and xo and p values for various solvents were calculated by using eq 10. With the ethanol-water curve being nonlinear, the data were fit to two separate straight lines in the solvent concentration ranges of 0.0 to 0.1 and 0.1 to 0.4. The xo and p values were calculated for each region, and model calculations for the two solvent concentration ranges were done by using respective xo and p values. By use of the description in ref 15, g12 is calculated from xlz. Water-Polymer Interaction Parameter, X23. The water-polymer interaction parameter was determined from the swelling of polymer in water at ambient temperature. The polymer latex was dried and thin films of the dry polymer were compression molded in a Carver press. The films were further dried in a vacuum oven at (28) Othmer, D. F.; Benenti, R. F. Ind. Eng. Chem. 1946 (March),299.
Figure 3. Chi parameter versus solvent volume fraction for MEK-water system. 60 O C for 24 h. Small film specimens were placed in water for a minimum of 7 days before measuring the water uptake by the film. The water-polymer interaction parameter, ~ 2 3 was , determined by using the equation
where $2 and $3 are the volume fractions of water and polymer in the swelled film, also $2 + $3 = 1and X 2 and X 3 are the molar volumes of water and the polymer, respectively. The process of water uptake and removal was repeated many times and was found to be completely reversible. PS film showed no swelling, indicating practically no interaction between PS and water. The water swelling of the acrylate terpolymer gave a X 2 3 of 2.1. Interfacial Tension Values. A calculation of individual terms in the equilibrium equations (eqs 5 and 6) shows that in the case of water-soluble solvents, the major contributions to thermodynamic equations come from terms corresponding to the water and the solvent. The Laplace term describing the interfacial tension effect is very small (less than 1%). Therefore, the choice of interfacial tension value is not critical to the calculation of solvent partition. The interfacial tension values for various systems in the literature vary from 1 to 30 mN/ m.1t3v9J4 A value of 20 mN/m was used in our calculations. The particle radius also appears in the Laplace term. With the Laplace term being relatively small, the particle size effect is also insignificant. This contrasts with solvents that are insoluble or have low solubility in water, where particle radius is a key factor in determining the solvent partition behavior.11J4 Other Parameters. The molar volumes for water and solvents were taken from the CRC handbook or estimated based on solvent density.29 The molar volume for a polymer is large compared to that of a solvent or water. The particle radius was obtained by using quasi-elastic light scattering. The values of these parameters are shown in Tables I and I1 of ref 19. Comparison of Experimental Data a n d Model Predictions The solvent partitioning data presented in a previous paper19 is compared with the BGDP model predictions. _____
~
~~
(29) CRC Handbook of Solubility Parameters and Other Cohesion
Parameters; Barton, A. F. M., Ed.;CRC Press: Boca Raton, FL, 1983.
Popli
76 Langmuir, Vol. 7, No. I, 1991
0.14
w
0
I
I: 0 J
a
0.06-
V
a
-0.02
0.000
0.010
0.020
0.030
0.0
0.040
SOL.VOL.FRAC.:AQ.PHASE
1 W
VI
o s
(e
0.2
SOL.VOL.FRAC.:
Figure 4. Solvent volume fraction in polymer phase versus in aqueous phase for DEGBA-acrylate terpolymer latex: ( 0 )
experimentaldata; (-) ternarycurves;
0.1
1
..) pseudobinarycurves.
0.4
0.3 AQ.PHASE
Figure 6. Solvent volume fraction in polymer phase versus in
aqueous phase for ethanol-acrylate terpolymer latex: ( 0 )experiment data; (-) ternary curves; (. .) pseudobinary curves.
-
-0 I
0.3.-
I
4
J I
0
n
u 0.2< a
U
5
I O
2.R
-
f
J
90
-I 0
>
dVI
0
V
0-
.
2 0
0.1-
0.00.00
0.20
0.10
1
SOL.VOL.FRAC.: AQ.PHASE
Figure 5. Solvent volume fraction in polymer phase versus in aqueous phase for MEK-acrylate terpolymer latex: ( 0 )exper-
imental data; (-) ternary curves; (.
-) pseudobinary curves.
The data along with the ternary or the pseudobinary model predictions are plotted for various solvents and two latexes in Figures 4 through 11. The solvent-polymer interaction parameter, ~ 1 3is, the adjustable parameter and is assumed to be independent of the solvent concentration. Acrylate Terpolymer Latex. No single curve fits the experimental data for solvent DEGBA, Figure 4. For the ternary model, a X13 value of -1.0 gives a good fit at low solvent concentrations. However, at the highest solvent concentration examined, the experimental data point is located on the curve corresponding to a x i 3 of 0.5. We have no independent data on the solvent-polymer mixture to guide us in selection of an appropriate X 1 3 . A similar behavior is observed for the pseudobinary model predictions. In contrast to the solvent DEGBA, the experimental results for MEK, Figure 5, closely follow the ternary and the pseudobinary model curves correspondingto X l 3 values of 1.0 and 0.0, respectively. Again, no independent measure of X13 is available for the MEK-acrylate terpolymer system. Ethanol partitions slightly to the polymer phase at high solvent concentration, Figure 6. The ternary or the
-
0.0
0.1
0 :2
SOL.VOL.FRAC:
0.3
0.4
AQ.PHASE
Figure 7. Solvent volume fraction in polymer phase versus in aqueous phase for DEGEE-acrylate terpolymer latex: ( 0 ) experimental data; (-) ternary curves; and * .) pseudobinary curves. (e
pseudobinary model give a reasonable fit of the experimental data for x i 3 values in the range of 1.5-2.0. The model predictions show little sensitivity to changes in X13; therefore a range of x i 3 values rather than a unique value is assigned by the curve fitting method. The behavior of solvent DEGEE, Figure 7, is very similar to that of ethanol. No partition of the solvent to the polymer phase is observed. A x i 3 value of 1.5 to 2.0, predicts partitioning behavior in agreement with the experimental data. The solvent-water interaction parameters are not available for DPGME; therefore, no attempt was made to fit the data in Figure 8. However, these results are quite similar to that of ethanol. PS Latex. The results for the PS latex were analyzed in a manner similar to that of the acrylate terpolymer latex. Polystyrene does not uptake water; therefore only the pseudobinary model was used for model predictions. The results for DEGEE, Figure 9, show no partitioning in the polymer phase and are best described by a high chi value in the neighborhood of 1.5-2.0. The results for ethanol and DPGME, Figures 10 and 11, show an increased partitioning of the solvent in the polymer phase relative
Langmuir, Vol. 7, No. 1, 1991 77
Swelling of Latex Particles I --'.- .--T 3
I -
-
-'
I
I
I
1
0.06
0.0
0.1
0.2
SOL.VOL.FRAC.:
a
+
0.1
0.0
0.3
t 0.3
0.2
S O L . V O L . F R A C . : AO.PHASE
A O . PHASE
Figure 8. Solvent volume fraction in polymer phase versus in aqueous phase for DPGME-acrylate terpolymer latex: ( 0 ) experimental data.
Figure 10. Solvent volume fraction in polymer phase versus in
aqueous phase for ethanol-polystyrene latex: ( 0 )experimental data; (. pseudobinary curves.
-
e)
w
a
n r T
do u
a 2.0 ......
-1
0
. l
0
Ln
I
0.0
0.1
0.2
SOL.VOL.FRAC.:
0.3
C
0.0
0.1
AO.PHASE
0.2
SOL.VOL.FRAC.:
0.3
(
4
AG.PHASE
Figure 9. Solvent volume fraction in polymer phase versus in aqueous phase for DEGEE-polystyrene latex: ( 0 )experimental data; pseudobinary curves.
Figure 11. Solvent volume fraction in polymer phase versus in aqueous phase for DPGME-polystyrene latex: ( 0 )experimental data.
to the case of acrylate terpolymer latex. The ethanol data fit is obtained for a xi3 of -1.5. A xi3 of -1.5 is rather low for ethanol-PS interaction consideringthat ethanol hardly swells PS film. The model predictions even for this xi3 value show a poor fit at high ethanol concentration. The data fit for DPGME was not attempted since water-solvent interaction parameters are not known for this system. In the previous paragraphs, we have compared the solvent partition curves predicted by the two models and the experimental data. Another test of the BGDP model would be an assessment of the relative order for the adjustable chi parameters values, ~ 1 3 that , provide the best fit to experimental data for various solvents. These values are shown in Table 11. Considering the acrylate terpolymer latex, the best fit of the experimental data for solvents DEGBA, MEK, ethanol, and DEGEE is obtained for correspondingly increasing chi values. The relative ordering of the chi parameter, ~ 1 3 ,is quite reasonable considering the solubility/swelling of the dry polymer in these solvents. The data are not sufficient to allow a similar comparison for the PS latex. However, the chi value for ethanol is rather low considering that ethanol shows only a slight swelling of the dry PS.
Table 11.
(e
e)
XIS
Values for Best Fit of the Experimental Data X13
solvent DEGBA
MEK ethanol DEGEE
ternary model -1.0 to 0.5 1.0 1.5 2.0
value pseudobinary model -1.5 to 0 0.0 to 0.5 1.0 >2.0
PS Colloid solvent DEGEE ethanol
XIS value for pseudobinary model
2.0
-1.5
In summary, the BGDP model has a strong theoretical basis and the assumptions made to derive eqs 5,6, and 7 or 8and 9 are reasonable. The model predictions, however, show only a qualitative agreement with the experimental data. To understand the cause for this discrepancy, we examine various sources of experimental error. As shown earlier, the experimental error of measuring the solvent partition is generally small.19 The partition measurements,
Popli
78 Langmuir, Vol. 7 , No. I, 1991
T
0
0.04
0.06
SOL.VOL.FRAC:
0.12
0.16
I
/
a 1
0:10 SOL.VOL.FRAC.:
Fi ure 12. BGDP model prediction of the effect of waterpofymer interaction parameter on the partition behavior for solvent MEK and acrylate terpolymer latex: (-) ~ 2 =3 2.0, (- - -) x a = 2.1, (. .) x23 = 2.2.
*,,=-1.o
0".00
A0.PHASE
t
a
': 0.4
0
0.20 AO.PHASE
Figure 14. BGDP model prediction of the effect of water-solvent interaction parameter on the partitioning behavior for solvent MEK and acrylate terpolymer latex: XIS = 1.0. In a similar manner, we examine the influence of errors in water-solvent interaction parameter, ~ 1 2 on , the predicted partition behavior. The experimental standard deviation of head space measurement is approximately 5-10% of the average value. This error carried through Flory-Huggin's equation, eq 11, leads to a standard deviation of 3-7 % for the calculated chi value. Another way to look at these errors is to consider the standard deviation of slope and intercept for aleast-squares straight line fit to a plot of solvent-water chi value versus the solvent concentration. We have employed the latter technique for estimating errors. Considering the concentration dependence of solvent-water interaction parameter XI2
= xo(1 + PV'J = xo + PXO - P X O V l l
standard deviations of approximately 2-5% and 3-10% of the average values are obtained for the intercept, xo + pxo, and the slope, pxo, respectively. The errors for g 1 2 are of similar magnitude SOL.VOL.FRAC:
AQ.PHASE
Figure 13. BGDP model prediction of the effect of waterpolymer interaction parameter on the partition behavior for solvent DEGBAand acrylateterpolymer latex: (-1 x23 = 2.0, (- - -1 X23 = 2.1, and X23 = 2.2. (e.
e)
however, are independent of the experimental determination of the interaction parameters x 1 2 and X 2 3 . The errors associated with measurement of the interaction , ~ 2 are 3 reflected in the predictions parameters x 1 2 , ~ 1 3and of the BGDP model. With the solvent-polymer interaction parameter, ~ 1 3being , a fitting parameter, no experimental error is associated with ita value. However, assumption of a solvent-polymer interaction parameter independent of the solvent concentration is not true in general. This assumption may be of concern, particularly for solvents that partition strongly in the polymer phase. In the following discussion, we will focus on the effect of errors in x 1 2 and X23 on the BGDP model predictions. First we consider the effect of the water-polymer interaction parameter, ~ 2 3 . Repeated measurements of the swelling of polymer by water gave a X 2 3 value of 2.1 f 0.1. BGDP model predictions for the solvents MEK and DEGBA are shown in Figures 12 and 13. It is clear that within the limits of experimental error, the model predictions are independent of changes in the water-polymer interaction parameter.
In Figures 14 through 17, we show the effect of varying slope and intercept values on the predicted partition for the acrylate terpolymer latex. The BGDP model predictions are very sensitive to changes in g 1 2 . For instance, a comparison of the model predictions for MEK and acrylate terpolymer latex shows that a 5 % change in slope is almost equivalent to changing the fitting parameter X 1 3 from 0.5 to 1.0, Figure 14. Similarly a 2% decrease in the intercept is equivalent to changing X 1 3 by 0.5. The sensitivity to changes in the g 1 2 value is even stronger for the solvent DEGBA, Figure 15. It must be pointed out that the predicted sensitivity to changes in g 1 2 values is also influenced by the X 1 3 value used in calculations. In general, the model predictions are less sensitive at high x i 3 value. The x i 3 value that gave the best fit to solvent partitioning data in Figures 4 through 10, was used for estimating the uncertainties of model predictions. In the case of solvents such as DEGEE and ethanol where the model predictions change only slightly upon varying X 1 3 , the model calculations also show a relatively weak effect due to changes in g 1 2 . However, the uncertainty of model prediction still remains because a unique ~ 1 3as , discussed earlier, cannot be assigned by the curve fitting method in these cases.
Swelling of Latex Particles
Langmuir, Vol. 7,No. 1, 1991 79
812 =
I
-
3.15
- 0.75
v',
w
ffl
a 0 a
z _I
0
n
.:
0
U
U
5 _I
0
>
$ 0 0)
0.04
0.00
0.02
0.01
0.03
0
4
0.0
SOL.VOL.FRAC.: AQ.PHASE
.....
m
.........
I
0.3
0.4
Figure 17. BGDP model prediction of the effect of water-solvent interaction parameter on the partition behavior for solvent ethanol and polystyrene latex: X13 = -1.5.
=
w
0.2
SOL.VOL.FRAC.: A Q . P H A S E
Figure 15. BGDP modelprediction of the effect of water-solvent interaction parameter on the partition behavior for solvent DEGBA and acrylate terpolymer latex: XI3 = -1.0.
B12
0.1
6.51 6.18
6.51
. 1.17 . 1.17 . 1.11
V'2
V'2 V.2
n 0.010 -I
a U
a
LL
2l 0.005 -I
0
ffl
0.000 0
7.3 0.1
0 :2
0
:a
0.00
0 :4
0.02
0.03
c
0.04
SOL.VOL.FRAC.
SOL.VOL.FRAC.: AO. PHASE
Figure 18. Chi parameter versus solvent volume fraction for aniline-water system (ref 16).
Figure 16. BGDP model prediction of the effect of water-solvent interaction parameter on the partition behavior for solvent DEGEE and acrylate terpolymer latex: XI3 = 2.0. The pseudobinary model shows a similar response to changes in g12 value. The extreme sensitivity of model predictions to small variation in the solvent-water interaction parameter is not restricted to the solvents and latexes used in this study. We examined the case of solvent aniline and cellulose acetate latex to illustrate this behavior. This system had been studied in detail earlier.l6 The parameter values needed to perform calculations for this system are taken from the previous study.16 The experimentallydetermined water-aniline chi values, ~ 1 2taken , from Table I1 of ref 16 are reproduced in Figure 18. The slope and intercept values have standard deviations of approximately 10% and 296, respectively. BGDP model calculations, for the aniline-cellulose acetate latex system, Figure 19, show a large variation in the predicted partition behavior due to 2-576 change in the slope or the intercept. Summarizing, in principle the BGDP model assigns a unique xi3 value that fits the experimental data. In practice, however, a large uncertainty in the choice of ~ 1 results due to the experimental error of measuringglz even when this error is relatively small. It must be made clear
0.01
3
that the results reflect not as much the weakness of the BGDP model but suggest that lack of very precise knowledge with regards to the interaction parameters is a major stumbling block in this type of calculation. The BGDP model also predicts the equilibrium amount of water in the polymer phase. However, no experimental method is known that can determine directly the water volume fraction of the polymer phase. BGDP model calculations for the cellulose acetate latex suggest that the water volume fraction in the polymer phase first increases and then decreases with increasing solvent concentration in the aqueous phase.16 The calculations for the acrylate terpolymer latex with various solvents, however, suggest no such universal behavior. In many cases, the water volume fraction in the polymer phase increases continuously with the increase in solvent concentration of the aqueous phase. The physical significance of these observations is not clear a t this time. We believe that a cautious interpretation of these predictions along with some experimental verification is desirable. Other issues that deserve attention in regards to the modeling of the partition data are as follows:
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80 Langmuir, Vol. 7,No. 1, 1991
i W
2 0.3 r
4
I
O
Q
u 0.2 d
Ct
LL -I
0
d
0.1
II)
0.04 0
0.01
0.02
0.03
0
4
SOL.VOL.FAAC.: AQ.PHASE
Figure 19. BGDP model prediction of the effect of water-solvent interaction parameter on the partition behavior for solvent aniline and cellulose acetate latex: x13 = 0.4.
(a) Prausnitz’s work suggests that solvent can be adsorped by the p0lymer.3~ Experimentally, the adsorbed solvent will be measured to belong to the polymer phase; however, the model in its present form predicts behavior based on solvent absorption considerations only. (b) The surfactants, at a concentration above the cmc, form micelles that have been shown to incorporate solvent from the aqueous pha~e.311~~ In the present study, the free surfactant was removed from the latex by dialysis. Therefore, this potential source of error has no bearing on the analysis here. (c) Lastly, the observations of Winnik question the traditional concept of particle surface area.33 They suggest that a latex particle morphology may be not unlike that of a sponge with micropores running through the particle. Therefore, the surface area of a particle would be many times that of a spherical particle and would depend on accessibility of the micropores to solvent or monomer of interest. Nevertheless, the Laplace term being small for water-soluble solvents, the effect on solvent partition may not be significant. A detailed understanding of these phenomena, such as adsorption and latex particle morphology,is needed before these factors can be included to develop a more comprehensive picture of the swelling of latex particles by solvents. (30)Bonner, D. C.; Prausnitz, J. M. J. Polym. Sci., Phys. Ed. 1974,12, 51. (31) Shih, L. B.; Williams, R. W. J. Phys. Chem. 1986,90, 1615. (32) Kaneshina, S.; Kamaya, H.; Veda, I. Biochim. Biophys. Acta 1984, 75, 777. (33) Winnik, M. A. Polym. Eng. Sci. 1984, 24, 87.
Conclusions The BGDP model provides a good theoretical basis for understanding the solvent partition behavior in a polymer latex. One of the main advantages of this model is ita ability to ascertain the relative importance of various parameters and interactions in governing the partition behavior. For example, Laplace term, which is of prime importance to the partitioning of hydrophobic solvents, is shown to be insignificant for water-soluble solvents. A comparison of the experimental partition data with the BGDP model predictions shows reasonable agreement, as shown here and in a previous study.16 The model predictions show a strong dependence on the solventwater interaction parameter. In general, the solvent partition behavior is best described by a range of x 1 3 values and not a unique value. An interesting prediction of this model is the presence and changes in the amount of water present in the polymer phase. The experimental verification of these predictions and its exploitation for applications remain interesting challenges. Acknowledgment. We thank Dr. C. Bindschaedler for his help to numerically solve the equations and Mrs. Anna Perkins for typing this manuscript. Appendix A solution of the equilibrium equations (eqs 5,6, and 7) is obtained by using the subroutine ZSPOW, a Fortran program to solve nonlinear equations, from the IMSL library of subroutines on the VAX system. This subroutine solves simultaneous equations by using an interactive procedure. The chi values for waterpolymer, an equation describing the water-solvent dependence on the solvent concentration, and avalue chosen for the solvent-polymer are some of the inputs needed for solving the equations. Other inputs include the relative ratios of the molar volume of solvent-water and polymerwater, the radius of the unswollen polymer latex particle, and the interfacial tension value. The right-hand side of the equations is evaluated upon selecting a value for the solvent fraction in the aqueous phase. The computer iteratively assigns values to the solvent, water, and polymer fraction in the polymer phase on the left-hand side of the equations, so as to obtain a least-squares fit. An important point to remember is that the solvent to water ratios in the aqueous and the polymer phases are not the same. Therefore, the chi parameter for solvent-water interaction being concentration dependent must be evaluated separately for two sides of the equation. A copy of the program for use on VAX system may be obtained from the author. Registry No. PS,9003-53-6.
