Langmuir 1992,8, 2210-2214
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Structure Factor for Colloidal Dispersions. Use of Exact Potentials in Random Phase Approximation V. K. Kelkar,+Janaky Nmayanan,i and C . Manohm**+ Chemistry Division, Bhabha Atomic Research Centre, Trombay, Bombay 400085, India, and Physics Department, R. J . College, Ghatkopar, Bombay 400086, India Received March 1 1 , 1992. In Final Form: June 16,1992 An exact DLVO potential consisting of hard core repulsion, Hamaker attraction, and Debye-Huckel electrostatic terms is used for the calculation of structure factor of colloidal dispersions in the random phase approximation (RPA) with hard sphere (HS) as reference system. The Fourier transform of the attractive term is calculated numerically while that of electrostatic term is evaluated analytically. The theory is then applied for a rich variety of micellar solutions such as pure nonionica (Cas and Triton X-lOO), mixed micelles (Triton X-1WSDS and Triton X-lWSA), and ionic micellar solutions showing clouding behavior (C&JBuSBr). The resulta are compared with earlier results on these systems which make use of mean spherical approximation and random phase approximation with the sticky hard sphere referencesystem. The present model reproducesall the salient features of previous theories and ie superior in giving correct order of magnitude of the Hamaker constant without any approximations in the form of the potential.
Introduction The structure and interparticle interactions in colloidal dispersions are interesting topics of investigation. There are various schemes which make use of different models for the interaction potentials and different liquid state theories to calculatethe structure factor SCQ).The choice of the form of the potential is guided by ease in solving the Ornstein-Zernike equation analytically. This has resulted mainly in using either squarewell potentials (hardsphere, sticky hard sphere) or Yukawa type potentials (screened Coulomb, short range attractive). This has also resulted in a number of anomalies. For example while use of screened Coulomb potential in mean spherical approxby Hayter-Penfold' neglects the attractive imation (MSA) contribution completely, the use of Yukawa type attractive potential by Hayter and ZulaufZ for clouding in nonionic micellar solution leads to unreasonable values of the strengths of interaction. Recently we3-5have applied the sticky hard sphere model to nonionic surfactant solutions. The model accounted for a number of features such as phase diagram, small angle neutron scattering (SANS), and light scattering (LS).The model also yielded considerable low values for the strength of the attractive potential (-2.5 kT) compared to that of MSA type theory (- 15 kT). We6 have also extended our model using the random phase approximation (RPA) for mixed ionicnonionic micellar solutions such as Triton X-100-SDS (sodium dodecyl sulfate) and Triton X-100-SA (salicylic acid). The model was also used7 to account for the SANS data on pure ionic micellar solutionsof &NBu3B+ which show clouding behavior.
In the present investigationwe have used amore realistic potential between spherical particles as in the DLVO scheme. The structure factor S(Q)is calculated in RPA with hard-sphere as the reference system. The Fourier transform of the attractive Hamaker term is calculated numerically while that of the electrostatic Debye-Huckel term is evaluated analytically as in ref 6. The theory is then applied to a number of micellar solutions. Data on nonionic, mixed ionie-nonionic, and ionic micellar solutions showing cloud point has been explained quantitatively. Our approach, though similar to that of Grimson? differs in evaluation of Fourier transform of the attractive and repulsive terms as also in using the form for the attractive potential. Grimson had used (llrS)approximationfor the attractive part and Fourier transform (FT)was evaluated for a s m a l l Q region only. He had used the treatment to account for phase separation in colloidal dispersionswith change in electrolyte concentration. We have used the exact Hamaker potential and the FT is evaluated (though numerically) over the entire Q range. In the present investigationwe have used our schemeto account for SANS and LS data. Thus our model can be considered as an extension of that of Grimsong with realistic potential and to wide Q range allowing the interpretation of scattering data on colloidal systems.
Theory The interactionpotential between two sphericalcolloidal particles separated by distance r is given by
V(r)= v h , + V,(r) + VJr)
(1)
+ Bhabha Atomic Research Centre. where v h c represents hard core repulsion and accounts for t R. J. Colleee. the finite size of particle. It is given by (1)Hayter, JYB.; Penfold, J. Mol. Phys. 1981,42, 109. (2)Hayter, J. B.; Zulauf, M. Colloid Polym. Sci 1982,260,1023. (3)Menon, S.V. G.; Kelkar, V. K.; Manohar, C. Phys. Rev. A 1991,43, V,, = for r Ia 1130. = 0 for r > a (4) Srinivaaa Rao, K.; Goyal, P. S., Dasannacharya, B. A.; Menon, S. (2) V.G.;Kelkar,V.K.;Manohar,C.;Miahra,B.K.PhysicaB1991,174,190. where a denotes the particle diameter. The second term (5) Srinivasa b o , K.; Goyal, P. S.; Dasannacharya, B. A.; Kelkar, V. K.; Manohar, C.; Menon, S. V. G. Pramanu 1991,37,311. in eq 1represents the van der Waals attraction term, the (6)Kelkar,V.K.;Mishra,B.K.;SrinivasaRao,K.;Goyal,P.S.;Manohar, C. Phys. Rev. A 1991,44,8421. (7) Warr, G. G.; Zemb, T. N.; Drifford, M. J . Phys. Chem. 1990.94. (9) Grimson, M. J. J. Chem. SOC.,Faraday Tram. 2 1983, 79,817. 3086. Grimson, M. J.; Laughlin, I. L.; Stilbert, M. J. Phys.: Condem. Matter (8)Manohar, C.;Kelkar, V. K. Langmuir 1992,8,18. 1991,3,7995.
0743-7463/92/2408-2210$03.OO/0 0 1992 American Chemical Society
Structure Factor for Colloidal Dispersions
Langmuir, Vol. 8, No. 9,1992 2211
expression for which for spherical particles has been given by Hamakerlo = -(A/12)[a2/(r2 - a') +a2/r2+ 2 In ((r' - a2)/r2]l (3) The various approximations for this term and the errors involved have been discussed by 0verbeek.l' For large W , and this approximation separations (r >> a) V, has been used by Grimsong for evaluation of structure factors for small Q regions. The third term represents electrostatic interaction between the colloidal particles and is well-known in Debye-Huckel theory''
V,(r)
-
V,(r) = f i e 2 exp[-K(r - u ) I / [ ~ (+ I 0.5~a)'rl (4) where Z is the charge on the colloidal particle, e the dielectric constant of the dispersion medium, and K is the inverse Debye screening length which is related to ionic strength (I)of the solution by
@ = 87rNAe21/(1000ek8T)
(5) In the present investigation we have used hard sphere as our reference system and the zeroth order approximation structure factor So(Q)has been evaluated by using empty core model13
I
CBE5 RPA-HS (HAMAKER)
3t \ 0.05
0.00
0.10
0 .I5
P (A'-')
Figure 1. SANS intensityI(Q) in arbitrary unite (ref 2) plotted a~ a function of Q for different temperatures: 0,327.6 (0.769); 0,323.1 (0.702);A,317.1 (0.512);A,311.1 (0.405);0,298.1 (0.109). The numbersin parenthesesindicate the (AlkeT)values required to fit.
The scattered neutron intensity is a function of both the particle form factorP(Q)and the interparticlestructure factor SCQ). In general it is not possible to separate these patrs and only in the case of spherical particles it can be written as17y7
l/So(Q) = 1 - 1 2 d I d 3- )'1 - 2)/(1- d41i1(Qa)/(Q4(6) where jl is the first-order spherical Bessel function and 7 is the volume fraction of particles which is related to the number density p by D = 7ra3p/6. In the random phase appr~ximationl~ the structure factor for our real system can be written as
SCQ)= So(Q)/[l+ (p/kBT)(V,(Q) + Vc(Q>]l (7) where V(Q)'s represent the Fourier transforms of the corresponding perturbation potentials. Vc(Q) can be evaluated analytically and is given by Vc(Q) = 47r?e2[K sin Qcr + Q cos QaI/[e(l+ 0.5Ka)'Q(Q2 + @)I (8)
V,CQ>
= JgmV,(r)[sin Qr/(Qr)14*r2dr
(9)
is evaluated numerically by using the DOlAJF library routine from the NAG-LIBRARY15on a NORSK-DATA computer. The routine is based on Gauss 10-point and Kronrw 21-point nodes. The upper limit of integration was restricted to 3a since the integral value was not sensitive to the values beyond that limit. The lower limit was chosen to be a(1 l.E-13) after checking for the convergence of the integral.
+
Results and Discussion In this section we will present the results of the application of our model to e~periments~*5~~J6 on several different micellar solutions. ~~
~
(10) Hamaker, H. C. Physica 1937.4, 1058. (11) Overbek, J. Th. G. Pure Appl. Chem. 1980,52, 1151. (12) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of the lyophobic colloids; &vier: Amsterdam, 1948. (13) Croxton, C. A. Introduction to liquid state physics; John Wiley and Sons: New York, 1975; p 81. (14) Hansen, J. P.;McDonald, I. R. Theory of simple liquids, 2nd ed.; Academic Press: London, 1986. (16) FORTRAN Library Manual Mark 10, Vol. 2, NAGL-LIBRARY 1796/0; MKS. (16) Manohar, C.; Kelkar, V. K.; Mishra, B. K.; Rao, K. S.; Goyal, P. 5.;Dasannacharya, B. A. Chem. Phys. Lett. 1990,171,451.
where V, is the volume of the monomer, N is the aggregation number, Cbt is the total surfactant concen, is the monomer concentration and pi and ps tration, C are, respectively,the micellar and solvent scattering length densities. We have used the same procedure for analyzing our data as in our previous studies as well as that used in literature. Thus the form factor chosen is that of spherical micelles
P(Q)= [3(sin QR - QR cos QR)/(QR)312
(11)
where R is the radius of the micelle. The micellar radii have been found to be almost same as in previous studies and consistent with packing considerations based on the length and volume of the hydrophobic parts. Since the details of this part have been discussed earlier in appropriate references, those have not been included here. (1) Pure Nonionic Micellar Solutions. Figure 1 shows the fits to SANS data of Hayter and Zulauf on C&& at different tem eratures. The value of micellar diameter and only the Hamaker constant A was was fixed at 48 varied. (A/&BT)values were found to be ranging from 0.109 at 25.1 "C to 0.759 at 54.6 "C. It may be noted that the fits are in general quite good except for temperatures close to the critical value. Figure 2 shows a linear variation of A with temperature. The increase in attractive potential with rise in temperature is consistent with several investigations on the phenomenon of c l ~ u d i n g . ~ .It~ Jmay ~ be noted that (A/ k ~ T cis) approachingclose to 1in the present model which is a more satisfying aspect as compared to both the Baxter model (-2.5) and the Hayter and Zulauf results (-15). This linear increase in A with temperature is more appealing in contrast to the dependence of T , the Baxter parameter, on temperature3 because A is directly the
1293
(17) Chen, S. H.; Lin, T. L. In Methods of experimental physics; Academic Preea: New York, 1987; Vol. 23B, p 489. (18) Reatto, L.; Tau, M. Chem. Phys. Lett. 1984,108, 292.
Kelkar et al.
2212 Langmuir, Vol. 8,No.9, 1992
‘1
I
C8E5
-
v, (3
T
0: W
C-
56
8 % WEIGHT
RPA -HS(HAMAKER)
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-
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0 IO
0 25
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298.4K
I
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TRITON X - 1 0 0 RPA-HS (HAMAKER)
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Figure 3. Linear relation used in Figure 2 to predict light scattering resulta for different q values. Note the poor quality of fits at higher temperature and shift in peak position to higher q.
strength of V , while 7 depends in a more complicated way to the potential. Figure 3 shows the theoretical predictions and comparison of light scattering results on C&5 using the A values obtained from SANS data. It can be seen that while the fits are better for the lower three temperatures they are not so good at the highest temperature where the maximum is seen to be shifted to higher volume fraction. Grimsong has shown qc to be 0.1287 using some approximations in a very similar model as was pointed earlier. Our shift of maximum in I(0)is around that value. It may be required to use a scaling procedure with the reduced volume fraction as was done in previous studies. The second possible reason for the poor fits at temperatures close to critical value may be in FWA being a poor approximation in a critical region where the correlation effects may dominate. With this point in view we also tried a scaling procedure for both SANS and LS data on C a s . We found that while the SANS data can be fitted excellently, the LS data fits become very bad at higher temperatures. Use of the PY approximation for &(Q) instead of the empty core model also gave similar results. Hence we have used the unscaled procedure with empty core SOCQ)as reference system in all our studies. Figure 4 shows fits to SANS data on another nonionic system, viz., Triton X-100micellar solution at 8% by weight (127mM). The diameter of the micelle was fixed to 56 A5J6and only the Hamaker constant was varied. The fits are comparable with our earlier results and the (A/ ~B!Ovalues are also of the right order (0.374-0.683). Figure 5 shows the variation of Hamaker constant with temper-
X
a
‘t
290
300
310 320 TEMPERATURE ( K )
330
340
Figure 5. Relation between Hamaker constant A and temperature from the fits to SANS data (Figure 4).
ature. The linear relation observed is consistent with a similar result on C&5 as well as with the stickiness parameter 7 va 1/T in the Baxter model. Thus we conclude from these studies that the preeent formulation accounts for SANS and LS data on nonionic micellar solutions. The model is poor at close to critical temperature. (2) Mixed Nonionic-Ionic Micellar Solutions. In this section we have applied the model to two cases of mixed micellar solutions where Coulomb terms in perturbation become important. The first system coneista of 127mM Triton X-100micellar solutionfor which different amounts of ionic surfactant sodium dodecyl sulfate were added and SANS data were recorded. This results in slowly building up of the correlation peak at finite Q values. Previously we had analyzed these data by using the RPA extension of the Baxter model and found the behavior of charge condensation and transition from nonionic to ionic micelles. Figure 6 shows the fits obtained by using the present model with micellar radius, effective charge 2 on micelle, and Hamaker constant A as variable parameters. Table I summarizes the values of the parameters. As can
Structure Factor for Colloidal Dispersions
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Langmuir, Vol. 8, No. 9, 1992 2213
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T R X - 1 0 0 (127mM)+SA(20mM)
TRITON X - 1 0 0 (127mMl+SDS R P A - H S (HAMAKER)
tt.
64
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56
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h
r
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I
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a 0
4 -
16
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8
01
I
I
0 0.000
I
0.050
0.100
0!150
Q(i-')
Figure 7. Fits to SANS data on 8% Triton X-100 + 20 mM salicylic acid at different pH values (ref 16). The optimum parameters are given in Table 11. The ordinates have been shifted successively by 8 units for convenience of plotting.
SDS, mM 1.33 6.65 13.3 33.3 53.3
R, A 26.9 26.0 24.7 21.5 20.5
A
X
1014ergs 2.04 2.64 2.83 3.13 2.73
agg no.
z
84
0.90 4.0 5.0 5.99 6.74
80 72 54 53
be seen the fits are as good as for previous model and other parameters such as radius of micelle and charge are ergs) is almost identical. The A value (Aav= 2.8 X seen to be almost constant but higher than that for pure ergs) Triton X-100 at the same temperature (1.5 X indicative of cumulative effect of SDS and Triton X-100. The charge condensation effect is also observed as in the previous model and has not been shown separately. The second system studied is that of 127 mM Triton X-100 with 20 mM salicylic acid (SA). The pH of the solution was varied to tune16 the Coulomb and van der Waals potential terms. Here the fits were obtained by varying micellar radius, charge on micelle, and Hamaker constant. The fits are shown in Figure 7 and the parameter values are summarized in Table 11. The fits are comparable to those obtained by the RPA Baxter model6and are better than those obtained by using the MSA and Yukawa type potential.16 The strength of the attractive potential is also comparable to kBTc. The 2 values are comparable to the ones obtained in a previous model6 and show the same titration curve type behavior with pH showing a sudden rise a t pK values of SA (not shown here). The A values are found to be almost constant (2.5 X ergs) being fixed by amount of SA and Triton X-100. (3) Ionic Micellar Solutions Showing Clouding Behavior. In this section we present the results of our model to micellar solutions of ionic surfactant dodecyltributylammonium bromide (C12NBuaBr) which show cloud point phenomenon. I t is expected that the effect of attractive and repulsive potentials will be important in deciding the behavior of this system. This system is unusual in that the monomer concentration increases with
Table 11. Parameters Obtained from Fits for Triton X-100 + SA PH R,A A X 1014ergs am no. z 1.98 2.39 2.86 3.35 3.89 4.36 11.46
27.8 27.8 27.8 26.7 26.75 26.15 26.05
2.34 2.44 2.44 2.64 2.64 2.54 2.24
93 92 93 82 83 77 76
0.64 0.95 1.17 1.44 1.85 1.99 2.05
Table 111. Parameters Obtained from Fits for ClzNBusBr Cbt, M C,,M N A X 1014ergs a,A KU Z
wt %
1.05 8.01 16.7 34.3
0.024 0.184 0.384 0.789
0.005 0.04 0.07 0.16
33 33 29 26
7.5 2.7 1.4 1.1
35.0 35.4 33.9 32.7
1.03 3.22 4.30 6.27
10.8 16.3 14.6 14.6
total concentration7 unlike other cases where it remains almost constant (=critical micelle concentration, cmc). The values of micellar and monomer concentrations were used as given in ref 7, which were obtained from the method of moments. Figure 8 shows the fits to SANS data of ref 7 with aggregation number, Hamaker constant A, and effective charge 2 on micelle as three variational parameters. The corresponding physical quantities are summarized in Table 111. It is seen that the fits are as good as were obtained previously using the sticky hard sphere model with RPA8 and are definitely superior to those of ref 7, which made use of hypernetted chain (HNC) approximation with Hayter-Zulauf type So(Q)as starting point. 2 and Ka values are almost identical to the ones obtained by previous model.8 The rise in inverse Debye screening length, K,leads to attractive potential taking over repulsive Coulomb potential and hence to clouding phenomenon. Conclusions We have demonstrated in the present paper that a very simple model where the structure factor S(Q) has been evaluated by using the random phase approximation with
2214 Langmuir, Vol. 8, No. 9,1992 -.-
~
--
I
c I 2 N Bu3Br ( 8 . 0 I
C12NBuSBr ( 1 , O 5 *A I N 3 3 j Z - I I j A-7.5 X I 6 l 4 e r g s PA
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Kelkar et al.
0.6
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j
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0.2
0.0 0.00
0.05
0.10
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0.20
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6.
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E
0
0
4.
Y
2
0.0 0.00
0.05
0.10
0.15
0.20
0.25 25
Q ( A'-I)
Figure 8. Fits to SANS data on ClzNBQBr (ref 6). The optimum parameters are given in Table III.
hard sphere as reference system and realistic potentials is able to explain SANS data on a number of micellar solutions. The results are comparable to those obtained from more complicated approaches in literature. The model gives correct order of attractive potential and explains SANS data in most of the cases. The results of the present paper correspond to an extension of ref 9 to
realistic potentials and for all Q ranges and enable experimental comparison. The model is however not satisfactory for LS and SANS data close to critical temperatures.
Acknowledgment. We are grateful to Dr. S. V.G. Menon and Mr. K. Srinivasa Rao for helpful discussions.
