Langmuir 1992,8, 2885-2888
2885
Fluidlike Ordered Colloidal Suspensions: The Influence of Electrolytes on the Structure Factor S( &) W. Hirtl, H.Versmold,t and U. Wittig Institut fiir Physikalische Chemie, RWTH, 0-5100 Aachen, Federal Republic of Germany Received May 4, 1992 The influence of the electrolytes NaOH and NaCl on the structure factor S(Q) of fluidlike ordered polymer colloid suspensionsis investigated. The charge of the latex particles 2,was determinedpreviously by using S(Q) as indicator in a titration experiment. Statisticalmechanical calculations based on this 2, and the commonly used screened Coulomb potential lead to far too structured S(Q). In order to obtain a consistent theoretical descriptionof the experimental SCQ) at various ionic strenghts,an effective charge 2 ., and Debye screeningparameter K* of the particlesis calculated on the basis of the Poisson-BoltzmannCell (PBC) model. The resulting 2,. and K* values are used to calculate S(Q) via the rescaled mean spherical approximation (RMSA).The influence of the electrolytes NaOH and NaCl on the structure factor S(Q) can semiquantitatively be accounted for by using only experimentally determined input parameters. Possible improvements are discussed.
Introduction Considerable effort has been directed toward the understanding of the self-organization of supermolecular systems to form fluid-, glassy-, and crystallike ordered 13tates.l~~ In order to understand the structure of charge stabilizedcolloidalsystemsfrom first principles, one should start with the determination of the particle charge. Next, statistical mechanical methods would be necessary to determine the effective pair potential of mean force V(r). Finally, statistical mechanics or computer simulation techniques should be used to predict the structures. Despite the simplicity of this concept, each of the steps mentioned above is faced with severe difficultie~.~ In a previous paper we used the structure factor S(Q) of fluidlike ordered suspensions, which were titrated with NaOH and NaCl respectively,to obtain an estimate of the particle charge ZP4 In the present investigation we refer to this particle charge and try to understand the influence of electrolytes like NaOH and NaCl on the structure factor S(Q)quantitatively. Taken the particle charge 2, as known,a pair potential V(r) is necessary to proceed. For colloidal particles suspended in a screening medium, the following effective pair potential of mean force V(r) has frequently been ~ d ' v ~
V(r)= rtotru2$~ exp(-K(r - u))/r
(1)
r>u
Here, r is the center-to-center distance between two particles, $0 the surface potential, t o the permittivity of free space, trthe dielectric constant of the solvent medium, u the diameter of the particles, and K the Debye-Huckel screening parameter. In the limit of low ionic strength and low surface potential, $0 is related to the particle charge 2, as $0 = 2d(TcotrU(2 + K U ) ) (2) (l)Safran, S. A., Clark, N. A., Eds. Physics of Complex and Supermolecular Fluids; Wiley: New York, 1987. (2) Pieranski, P. Contemp. Phys. 1983,24,25. (3)Pusey, P. N. In Liquids, Freezing and the Glass Transition; Levesque, D., Hamen, J.-P., Zinn-Justin, J., Eds.; North-Holland Amsterdam, 1991. (4) Versmold, H.; Wittig, U.; HMl, W. J. Phys. Chem. 1991,95,9937.
The requirement of low surface potential $0 is often not met with polymer colloid particles. For the particles used in this investigation, we determined a particle charge 2, = 2040 e- previ~usly.~ With KU smaller than 1 and 2, = 2040 e-, one obtains from eq 2
= 800-900 mV which must be compared with $0 = 25 mV, the upper limit of validity of the Debye-Hiickel approximation. It is thus obvious that eqs 1and 2 cannot directly be applied. In order to maintain the convenient mathematical form of eq 1,a charge rescaling procedure, known as the PoissonBoltzmann-Cell (PBC) model, has been p r o p ~ s e d .For ~ dilute colloidal systems the interparticle distances are so large that the real potential approaches the screened Coulomb potential (Debye-Htickel) form in a region midway between any two particles. The PBC model starts from the assumption that each particle is confined to a spherical (Wigner-Seitz) cell, for which first the PoissonBoltzmann equation is solved numerically. Next, in the vicinity of the cell boundary, the equivalent Debye-Hiickel solution is determined. This defines an effective particle charge Zp*, an effective surface potential $o*, and an effective screening parameter K*. Starting from a particle charge 2, = 2040 e-/particle, the rescaled charge is expected to be in the range 2 ., = 400-600 e-/parti~le.~ Once an adequate potential is determined, severalliquidstate theories are availableto calculate the static structure factor S(Q).6 In this paper we refer to the work of Hayter and Penfold' who solved the Ornstein-Zernicke equation in the mean spherical approximation for the potential V(r) given in eq 1. They obtained closed analytic forms of the direct correlation function c(r) and the structure factor SCQ). One defect of the original treatment, which occurs at high dilution, has later been removed by Hansen and Hayter.8 We shall use their rescaled mean spherical approximation (RMSA) as a convenient tool to generate theoretical structure factors for the comparison with our experimental SCQ). (6) Alexander, S.;Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.
J. Chem. Phys. 1984,80,5776.
(6) Hamen, J. P.;McDonald,I. R. Theory of Simpleliquids;Academic Press: New York, 1976. (7) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (8) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982,46,651.
0743-7463/92/2408-2885$03.00/00 1992 American Chemical Society
Hiirtl et al.
2886 Langmuir, Vol. 8,No.12, 1992
Computation For simplicity, the polymer colloid particles are taken as monodisperse and spherical. The complex amplitude of the electric field, E, of N scattering particles, each of which contains n scattering centers with scattering amplitudes f , is (3)
where ri,(t) is the position vector of the scattering center a in particle i a t time t. Q is the scattering vector, lQI = ( 4 d X ) sin (8/2), with 8 the scattering angle and X the wavelength of light in the medium. If the position vector of the center of particle i a t time t is Ri(t), we can write
ria(t)= Ri(t) + bi,(t)
(4)
where bi&) is the position vector of the scattering center a with respect to the center of particle i. The timeaveraged intensity can be written as9J0
(Z(Q1) = n 2 f N p(Q)SCQ)
(5)
with the particle form factor
and the structure factor
or the pair distribution function g(r). Since the structure factor SCQ)is the primary quantity obtained from our experiments, the comparison of experiment with theory will be carried out in Q space in this paper.
Experimental Section The light scattering setup has been described in detail previously.1° Essentially, it consisted of a modified Malvem Instruments photon correlation spectrometer. Particular care was taken to adapt the original goniometer for the performance of reliable intensity measurementa. Further, the manual goniometer was replaced by a stepper-motor-drivenpreciaion goniometer, Model IT6 DCA manufactured by Micro Controle. Photon counting and data processing were achieved with a Malvem Instruments K 7026 correlator system. The colloid particles used in this investigation were spherical standard Dow polystyrene particles of nominal diameter u = 91 nm. Dynamic light scattering experiments performed with a highly dilute suspension gave the somewhat smaller particle diameter u = 80 nm. The surface charge of these particles is mainly due to-OSO~-groups.To achieve fluidlikeordered s t a t e , the suspensions were kept in contact with a mixture of ion exchange resins Amberlyst A-27 and Amberlyet 15, both purchased from Fluka, Switzerland. The ion exchange resins were carefully cleaned before use as described by Vanderhoff et al.lS The samplepreparation and light scattering experiments have been described in detail recently.' The experimentalconditions for one sample without additional electrolyte as well as for three samples titrated with NaOH and three samples titrated with NaCl are shown in Table I. Structure factors S(Q) were determined according to eq 9. For this, the scattering intensity (Z(Q)) of the ordered and, in a reference experiment, the intensity (Z(Q))o of a completely disordered sample were measured at various Q values.
~~~~~~
(9)Pusey, P. N.;Tough, R. J. A. In Dynamic Light Scattering; Pecora, R., Ed.;Plenum: New York, 1984. (10) Hiutl, W.; Versmold, H.; Wittig, U.; Marohn, V. Mol. Phys. 1983,
50, 815. (11)Hiutl, W.; Veremold, H. J. Chem. Phys. 1988,88, 7157.
(12) Belloni, L.; Drifford, M.; Turq, P. Chem. Phys. 1984,83, 147. (13) Vanderhoff, J. W.;VandenHul,H. J.;Tanek,R. J.M.;Overbeek,
J. Th.G. In Clean Surfaces: Their Reparation and Characterization for Interfacial Studies; Goldfinger, G.,Ed.; Marcel Dekker: New York, 1970.
3r
Langmuir, Vol. 8, No. 12, 1992 2887
Fluidlike Ordered Colloidal Suspensions
g 2 t Ln
2
n
8 10
n "
2
1
n
1 0 ~ ~ ~ - ~
Figure 1. Experimental (* * *) and calculated (-) S(Q) of sample 1: n, = 1.8 x 10l8particles/m3, no electrolyte added. 3
3
I
0
Q
/ 1
2
0
1
2
Q I 1 0 ~ ~ - '
0 ~ ~ - ~
Figure 2. Experimental (* * *) and calculated (-) S(Q): (a) sample 2a, np = 1.79 x 1018 particles/m3and 3.1 pmol/L NaOH added; (b)Sample2b,n,= 1.79 X 1018particles/m3and3.1 pmol/L NaCl added.
Y
1
2
n
0
u
0
c
f 4
0
Q 1 lo5--' Figure 3. Experimental (* * *) and calculated (-) S(Q): (a) sample 3a, np = 1.78 X 10l8particles/m3and 6.2 pmol/L NaOH added, (b)sample3b,n,= 1.78 X 1018particles/m3and6.2pmol/L NaCl added.
GI
In order to remove dust scattering,the sampleswere multipass filtered through membrane filters. All experimentswere carried out at 20 1 O C .
*
Rssults and Discussion The experimental structure factors SCQ) of sample 1 (without additional electrolyte) is shown in Figure 1. Similarly, experimental S(Q)s of pairs of samples, each one with a certain amount of added NaOH and one with the same amount of NaC1, are presented in Figures 2-4. The experimental conditions are summarized in Table I. First, we note that NaOH influences the structure factor S(Q) much less than NaC1. For example, an amount of 3.1 X lo* mol/L Na+ ions when added as NaOH has almost no effect on the structure, which can be seen by comparing Figure 2a with Figure 1. By contrast, the same amount of NaCl leads to a significant loss of structure as indicated by Figure 2b. This can be understood if one remembers that before the titration the latex particles were transferred into the protonated form. When titrated with NaOH, the H+ counterions of the particles form water with the OH-
1
2
0
1
2
1oScm-' Q 105--' Figure 4. Experimental (* * *) and calculated (-) S(Q): (a) sample 4a, n, = 1.77 X 10l8particles/m3and 9.2 pmol/L NaOH added; (b)sample4b, np= 1.77 X 1018particles/m3and 9.2pmoVL NaCl added. Q
ions until all H+ ions are used up. During this period of the titration the ionic strength does not change since H+ ions are merely exchanged by Na+ ions. In the case of titration with NaC1, no such neutralization takes place and therefore the ionic strength varies from the very beginning. This explains why NaOH is much less effective in destroying the fluidlike structure of the suspensions at the beginning of the titration than NaC1. This different behavior of NaOH as compared with NaCl and the previously determined particle charge, 2, = 2040 e-/particle,4 which determines the initial H+ ion concentration, have to be taken into account during the calculation of the Debye-Htickel screening parameter
In eq 12 ni is the number density of ionic species i and the summation extends over all small ions in the system. Next, we investigate whether the variation of S(Q) on addition of NaOH or NaCl can be understood quantitatively with the concept presented in the introduction. With the particle number densities npand the amounts of added NaOH or NaCl as given in Table I, PBC calculations were carried out. Results for the effective particle charge 2,. and the effective screening parameter in units of K*U are included in Table I. We call attention to the large reduction of the charge from the value 2, = 2040 e-/particle to the effective charge 2,. = 450-500 e-/particle. With the PBC results given in Table I, next, RMSA calculations were performed. The calculated S(Q) are shown in Figures 1-4 as solidly drawn lines. In view of the fact that we did not use any adjustable parameter in the calculations, the agreement between the experimental and theoretical S(Q) is remarkable and demonstrates that a t least a semiquantitative description of charge stabilized suspensions can be achieved with this approach. Although all basic features of the experimental S(Q) are reproduced by our combined PBC-RMSA calculations, certain systematic deviations between experiment and theory are apparent. First, we note that for the highly ordered samplesthe experimental S(Q)sare systematically larger a t low Q than the Calculated ones. This is due to polydispersity and/or dust scattering. Next, we consider the more serious diagreement which occurs at the first peak of the S(Q)s, the height of the experimental curves being higher than that of the calculated ones. This is most pronounced in Figure 1,somewhat less in Figure 2a, but, with the exception of Figure 2b, also visible for the other samples. The simplest way to account for this behavior would be to increase the particle charge. One could argue that the accuracy of our S(Q) particle charge titration experiment is not sufficiently well established. If, for example, the effective charge of sample 2a
H&rtl et al.
2888 Langmuir, VoE. 8,No. 12, 1992
E li
-t 8
0
1
'F 0 0
2
1
Q
(* * *) and calculated (-) S(Q) for sample 2a. Effective particle charge increased to the PBC model saturation value 2,. = 580 e-/particle of u = 91 nm particles. For details see text.
Figure 5. Experimental
is increased from Zp* = 503 to 580 e-/particle, an almost perfect agreement between the experimental and theoretical S(Q)s is obtained. The resulting S(Q)s are shown in Figure 5. Thus, the question is how high a true particle charge 2, is needed for the PBC model to generate the higher value Zp*= 580 e-/particle. PBC calculations show that the saturation value of the effective charge for particles with a diameter u = 80 nm at the particle number density of interest isZp*= 526 e-/particle, too low avalue to achieve perfect agreement. On the other hand, if it is assumed that the particle diameter is u = 91 nm (the nominal diameter), then the saturation value of the particle charge increases to Zp* = 581 e-/particle. Thus, a too small particle diameter determined by the dynamic light scattering experiment would explain the discrepancy between the experimentaland theoretical peak heights of the S(Q)s. A second perhaps more probable explanation for the just mentioned discrepancycould be a known difficulty of the RMSA. Comparisons of Monte Carlo with RMSA calculationshave s h o ~ n lthat * for the same screened Coulomb potential the RMSA systematically underestimates the height of the first peak of the structure factor. A second deficiency of the calculations concerns an increasingshift between the firstpeaks of the experimental and theoretical S(Q)swith increasing NaCl concentration. This behavior was observed in our laboratory previously.I0 As a possible explanation, we considered the dilution of the sample due to the iterative addition of NaCl solution. In fact, for the samples with high NaCl concentration, a significant improvement can be obtained if the particle number density is lowered. In Figure 6 this is demonstrated for sample 4b, for which the largest shift between the first peak of the experimental and theoretical S(Q)s ~
~~
(14) Svensson, B.; JBnsaon, B. Mol. Phys. 1983,50, 489.
2 5
-1
/ 10 cm Figure 6. Experimental (* * *) and calculated (-) S(Q) for sample 4b. RMSA calculation with particle number density lowered to np = 0.8 X lo1*particles/m*.
was observed. In order to obtain the almost perfect agreement shown in Figure 6, the particle concentration had to be lowered from n, = 1.77 X 10l8m-3 to 0.8 X lo1* m-3, i.e. by more than a factor of 2. Since such a large error in the experimental particle number density can be ruled out, we tend to attribute also this second deficiency to the RMSA model. Monte Carlo simulations would be very valuable to clarify whether our suspicions concerning the RMSA are justified or not.
Conclusion The influence of NaOH and NaCl on the structure factor S(Q) of fluidlike ordered colloidal suspensions is investigated by light scattering. The different response of the structure factor S(Q) when equal amounts of NaOH or NaCl are added to a suspension is attributed to an ion exchange mechanism, which takes place with NaOH but not with NaC1. The behavior indicates that the particles interact via diffuse double layer repulsion. By assuming (a) that the interactino between the particles can be accounted for by the screened Coulomb potential V(r)given by eq 1,(b) that the effective charge Zp* entering V(r)can be determined via the PBC model, and (c) that the fluidlike structure can be accounted for by the RMSA model, a semiquantitative description of the two series of titration experiments without any adjustable parameter has been achieved. Remaining differencesbetween the experimentaland calculated S(Qs appear to originate from the RMSA model. Monte Carlo simulations would be very useful to clarify this last point. Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. Registry No. NaOH, 1310-73-2;NaC1,7647-14-5; polystyrene, 9003-53-6.
