- Email: [email protected]
Small angle light scattering studies concerning aggregation processes
246
Small angle light scattering studies concerning aggregation processes Daniela Asnaghi, Marina Carpineti, Marzio Giglio* and Alberto Vailati Substantial work, both experimental and theoretical, has been performed on aggregation processes in dense solutions. Aggregation driven by depletion interactions has recently been studied in a variety of systems. The phase diagrams that have been obtained are extremely complex, and they include aggregation and gelation phenomena.
Addresses Dipartimento di Fisica e Istituto Nazionale per la Fisica della Materia (INFM), via Celoria 16, 20133 Milano, Italy "e-mail: [email protected] Current Opinion in Colloid & Interface Science 1997, 2:246-250 Electronic identifier: 1359-0294-002-00246 © Current Chemistry Ltd ISSN 1359-0294 Abbreviations OLCA diffusion-limited cluster aggregation OWS diffusing-wave spectroscopy RLCA reaction-limited cluster aggregation SALS small angle light scattering SANS small angle neutron scattering SAXS small angle X-ray scattering
more biological interest will also be discussed. Results in the two year period under scrutiny here will then be presented. Recent results concerning dense solutions will be described in the second section. These studies have unveiled interesting analogies between colloids and other more general thermodynamic systems. We will cover in the third section aggregation phenomena and low angle scattering from depletion interaction colloidal systems. We will not cover work done with diffusing-wave spectroscopy (DWS), in spite of the fact that it deals with scattering of light, often collected at very small angles. DWS for its own nature yields information on small length scales, and this is at variance with the technique we deal with here (DWS is discussed by Grier (pp 264-270) in another review in this issue of Current Opinion in Colloid & Interface Science). We will take, however, the liberty of quoting low angle work done at other wavelengths (neutron or X-ray) if the matter under discussion is very close to that highlighted here in conjunction with SALS.
Introduction
Fractals and universality in dilute solutions
Aggregation phenomena are fairly ubiquitous and they appear in a vast number of physico-chemical systems. Although they have been a classical topic in physical chemistry for many years, interest in them picked up momentum in the late eighties when it was realized that colloidal aggregation leads to fractal structures and that aggregation processes are endowed with universal properties.
The early works were in connection with irreversible colloidal aggregation in dilute solutions. A very important achievement was the discovery that there are two limiting universal classes of aggregation kinetics [1], namely the diffusion-limited cluster aggregation (DLCA) [2] and the reaction-limited cluster aggregation (RLCA) [3]. The former occurs when the growth is fast and controlled by pure diffusive motion, and the latter when the growth is hindered by a (weak) repulsion which makes necessary many close encounters before a bond is formed. Both regimes are characterized by universal properties: aggregates' fractal dimension, reaction kinetics, and cluster size distribution. Ingenious analysis via scattering from individual clusters has been devised [4], and cluster size distributions for the different universal aggregation routes have been determined. The same technique has reappeared in recent work [5]. The intensity distribution as a function of the scattering wave vector q yields quantitative estimates of the fractal dimension df through the asymptotic, large q power law dependence I(q) cc q-df [6]. Also, the average gyration radius and mass can be obtained as a function of time, and this is valuable for the analysis of the reaction kinetics [7.].
The formation of large structures very naturally calls for techniques like light scattering, small angle static light scattering (SALS) in particular. SALS systems are miniature replicas of the small angle neutron and X-ray scattering systems (SANS and SAXS, respectively), the scattered radiation .being collected by multi-element sensors covering a narrow range of angles near the forward direction. This detection is often done directly, without image-forming optics, and a large fraction of the scattered signal is actually collected by the sensor. The added difficulty here is that diffraction and interference effects can severely affect the performance. Diffraction spilling from the main beam can add stray light at very small angles, and random interference of the scattered light (speckles) can introduce large statistical errors. We will cover very briefly in the first section the origins of the recent work on colloidal systems undergoing aggregation in dilute solutions, and a few studies of
Light scattering studies have qualitatively shown that protein solutions can also lead to aggregation processes [8,9]_ A fairly recent and thorough study that uses, among other techniques, light scattering and SANS has
Small angle light scattering studies Asnaghi et al.
shown quite convincingly that heat-induced denaturation of proteins in solution may generate clusters that fit very well into the RLCA scheme [10). A further study [11), with SANS alone, also investigated the intraparticle structure of clusters that originated in a heat set aggregation of proteins at a different pH . Once the clusters are diluted, they show a typical DLCA structure. At denser solutions, however, anticorrelation effects of the type described in the next paragraph are also reponed. Some evidence of salt-induced clustering in protein solutions (lysozirne) has also been presented [12).
Figure 1 70
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Aggregation in dense solutions Studies performed in dense colloidal solutions have evidenced intriguing similarities between irreversible processes in quite different systems. Whereas in dilute solutions, the scattered intensity as a function of angle decays monotonically, in dense solutions a peak at a finite wave vector was reported for reactions occurring in the OLGA mode [lSI (see Fig. O. Such a feature is typically observed during spinod al decomposition, the phase separation process of thermodynamically unstable systems [16]. Another intriguing fact is that the scattered intensity distributions I(q) at various times seem co possess the same scaling properties already found for the spinodal decomposition case: I(q/qm) =' q m-JF(q/qm), where qm is the peak position, and F(q/qm) is a time-independent scaling function. For ordinary spinodal decomposition d ~ 3, whereas in the present case d =' dr, where dr is the fractal dimension. A plot of the scaled intensity distributions is shown in Figure 2. The appearance of a peak in the scattered pattern was almost simultaneously reponed for a variety of systems, including semiconductor crystallites grown in glass matrices [17,18), crystallization of colloidal spheres (19), liquid emulsion aggregates [20), and colloidal aggregation in two dimensions [21). Recently, a peak in the scattered distribution has also been observed during the formation of gels, when gelation competes with two-phase separation processes [22,23). During spinodal decomposition, a bicontinuous structure of domains of alternating density is created, and it is this quasi-sinusoidal modulation that generates a peak at a finite wave vector in the scatte red pattern. Quite generally, the fact that I(q =0) =0 implies that the integral of the net density correlation function vanishes. For aggreg ation processes, this means that the clusters grow by feeding on a nearby depletion region over which mass is conserved. Computer simulations have reproduced these features both in two [24-26) and three dimensions [27-).
o
5000
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15000
20000
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30000
q (em") o 1997 Current Op inion in Conoid & Inlerlace Science
Intensity distributions as collected at various times during a OLGA process in a highly concentrated sample. Whereas in dilute solutions the intensity as a function of angle decays monotonically, in this case a peak at a finite q. moving in time towards zero, can be observed.
In the 3D case [27-], reaction-limited aggregation has also been simulated, and a finite-q peak in I(q) is found, at variance with preliminary experimental results [28]. Some works have tried to explain the presence of a peak in the scattered intensity and how it evolves in time. A phenomenological model based on mass conservation arguments [29-) has been proposed, and an analytical expression for the intensity distribution in good agreement with the experimental data has been found. Furthermore, a mean field model has been proposed, based on two equations which control the growth of the average cluster mass and the time-dependence of the cluster number concentration [30--,31--) . The predictions of this model are compared with experimental dat a [IS) and Brownian dynamics simulations and the agreement is excellent. According to this theory, the scaling in the later stages of aggregation is only apparent, as the rate of growth of the depletion region is slightly different from that of the growing cluster and the system is not describable in terms of a unique length. A correlation function (32), which predicts a scattering peak at finite wave vector, has been proposed and SAXS data are fitted with this new model. Finally, we point out that in dense solutions the aggregation eventually stops when the intercluster distance equals the average cluster diameter. This means that the aggregates form an interconnected network (colloidal gel) [33] which fills all the available volume.
248
Scattering and surface forces
that the interparticle potential proposed in [41] describes well the structure factor measured at various polymer concentrations by SANS.
Figure 2
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1997 Current Opinion in Colloid & Interlace Science
Pial of Ihe scaled curves I(q/qmlqmd f, where qm is Ihe peak position, and d t is the fractal dimension. The data refer 10 the later stages of the process shown in Figure 1. The intensity distributions have scaling properties similar to those found for the spinodal decomposition case.
It has been ascertained, with SALS and SAXS, that some porous media [34-36] exhibit features which are strongly similar to those of colloidal gels. In particular, they can be described as a matrix of closely packed fractal aggregates which retain their individual structure.
Aggregation and depletion interactions Recently, the attention of the colloid physics community has been gained by very interesting systems, namely colloid-polymer mixtures; there has been pioneering experimental work in the area that dates back to the early eighties [37-39]. The presence of the polymer gives rise to attractive 'depletion-induced' forces [40,41]. The origin of these forces is simple to understand qualitatively. Let us call the radius of the colloid R, and the polymer gyration radius rG. If the colloidal particles come to a distance d (skin-skin distance) smaller than rG, then there is a region between the particles into which the polymers cannot get. As a consequence, the osmotic pressure due to Brownian collisions on the outer regions is not balanced, and a net attractive force results. The system is extremely good to work with because the interparticle potential can be tailored at will by adjusting the range of the -attractive forces (controlled by rG), and the depth of the potential well (controlled by the polymer concentration and by the ratio rdR). In a recent paper [42°], it was shown
The phase diagram of colloid-polymer mixtures is exrrernely complex and can include virtually all the features of a monatomic fluid [43,44°°,45]. For small values of the ratio rcfR, one obtains fluid-crystal coexistence, whereas for larger values the phase diagram exhibits a fluid-fluid (vapour-liquid) boundary which can display a critical point and a rricritical point. A quantitative study of fluid-fluid phase separation kinetics with SALS has been recently presented [46°]. A peak at a finite angle, collapsing to q = 0 has been observed and discussed in terms of spinodal decomposition features. In addition to phase transitions of the type encountered with simple fluids, however, depletion interaction systems also display reversible aggregation and gelation. SALS again proves to be of great help in these studies. Experimental investigations [47-49,50°°] on a system with a small ratio, namely rdR =0.08, have shown chat, indeed, a ring appears and rapidly collapses to a stationary ring at very small angle, in a way very similar to that observed in aggregation phenomena in dense colloidal solutions [15]. At this point, the system attains a gel state which exhibits fractal morphology. This gel is transient and ultimately undergoes a liquid-solid separation with the associated onset of macroscopic sedimentation and disappearance of the scattering ring. The transition from a fractal gel state to a restructured configuration is very likely due to the fact that depletion forces naturally favour dense packing [51°]. Experimental studies on systems with different rcfR ratios have shown that gels with various morphologies can be obtained. Nonfractal gels have been observed in colloid-polymer mixtures with a large ratio (rcfR = 0.25) [52°], whereas a fractal gel, although characterized by a monotonously decaying scattered intensity distribution, has been found in mixtures with rcfR=0.03 [53].
Conclusions In the period considered in the present survey, little work has been done in the classical area of aggregation in dilute solutions. l\luch more interest has been attracted by aggregation in dense solutions, because of the added complexity due to intercluster interactions. Particularly interesting is the area of depletion-induced forces. These systems show a remarkable complexity of phase diagrams, and their experimental exploration is eased by the fact that interactions can be easily controlled. Aggregation and gelation processes show novel features, and it has emerged that studies in this area may lead to a better understanding of fundamental aggregation processes in biological samples, as the interactions involved are similar to those in depletion interaction systems.
Small angle light scattering studies Asnaghi et a/.
References and recommended reading
22.
Asnaghi 0, Giglio M, Bossi A. Righetti PG: large-scale microsegregation in polyacrylamide gels (spinodal gels), J Chem Phys 1995. 102:9736-9742.
23.
Tromp RH, Rennie AR, Jones RAL: Kinetics of the simultaneous phase separation and gelation in solutions of dextran and gelatin. Macromolecules 1995, 28:4129-4138.
24.
Sintes T, Toral R, Chackrabarti A: Dynamical scaling of fractal aggregates in dense colloidal solutions. Phys Rev £ 1994, 50:3330-3333.
25.
Earnshaw JC, Robinson OJ: Inter-cluster scaling in twodimensional colloidal aggregation. Physica A 1995,214:23-51.
26.
Earnshaw JC, Harrison MBJ, Robinson OJ: local order in two-dimensional colloidal aggregation. Phys Rev £ 1996, 53:6155-6163.
Papers of particular interest, published within the annual period of review, have been highlighted as: • ••
of special interest of outstanding interest
1.
Family F, Landau DP (Eds): Kinetics 01Aggregation and Gelation. Amsterdam: North-Holland; 1984.
2.
Lin MY, Lindsay HM, Weitz DA, Klein R, Ball RC, Meakin P: Universal diffusion-limited colloid aggregation_ J Phys Condens Matt 1990, 2:3093-3113.
3.
Lin MY, Lindsay HM, Weitz DA, Ball RC, Klein R, Meakin P: Universal reaction-limited colloid aggregation. Phys Rev A 1990,41:2005-2011.
4.
Broide ML, Cohen RJ: Experimental evidence of dynamic scaling in colloidal aggregation. Phys Rev Lett 1990, 64:2026-2029.
5.
Fernandez-Barbero A, Schmitt A, Cabrerizo-VilchezM, MartinezGarcia R: Cluster-size distribution in colloidal aggregation monitored by single-cluster light scattering. Physica A 1996, 230:53-74.
6.
Sinha SK: Scattering from fractal structures. Physica 0 1989, 38:310-314.
Asnaghi 0, Carpineti M, Giglio M, Vailati'A: light scattering studies of aggregation phenomena. Physica A 1995, 213:148-158. A review of the most interesting results in the field of colloidal aggregation is presented.
7.
8.
Patkowski A, Gulari E, Chu B: long range tRNA-tRNA electrostatic interactions in salt-free and low-salt tRNA solutions. J Chem Phys 1980, 73:4178-4184.
9.
Giordano R, Maisano G, Mallamace F, Micali N, Wanderlingh F: Structural properties of macromolecular solutions. J Chem Phys 1981, 75:4770-4775.
10.
Gimel J·C, Durand 0, Nicolai T: Structure and distribution of aggregates formed after heat-induced denaturation of globular proteins. Macromolecules 1994, 27:583-589.
11.
Renard 0, Axelos MAV, Boue F,Lefebvre J: 'Ordered' structure in solutions and gels of a globular protein as studied by small angle neutron scattering. Biopolymers 1996, 39:149-159.
12.
Minezaki Y, Niimura N, Ataka M, Katsura T: Small angle neutron scattering from lysozyme solutions in unsaturated and supersaturated states (SANS from lysozyme solutions). Biophys Chem 1996, 58:355-363.
13.
Broide ML, Garrabos Y, Beysens 0: Nonfractal colloidal aggregation. Phys Rev £ 1993, 47:3768-3771.
NarayananT, Petit J-M, Broide ML, Beysens 0: Desorptioninduced fragmentation of silica aggregates. Phys Rev £ 1995, 51 :4580-4584. Small angle light scattering is used for studying the dynamics of fragmenta· tion in reversible colloidal aggregation phenomena
14.
249
Gonzalez AE, Ramirez-Santiago G: Spatial ordering and structure factor scaling in the simulations of colloid aggregation. Phys Rev Lett 1995, 74:1238-1241. 3D simulations of colloid aggregation are presented. The appearance of a peak in the I(q) is explained in terms of density modulations. A finite-q peak from reaction-limited aggregation is also found.
27.
28.
Carpineti M, Giglio M: Transition from semiorder to disorder in the aggregation of dense colloidal solutions. Phys Rev Lett 1992, 70:3828-3831.
Carpineti M, Giglio M, Degiorgio V: Mass conservation and anticorrelation effects in the colloidal aggregation of dense solutions. Phys Rev £ 1995, 51 :590-596. A phenomenological model is presented to understand the 'spinodal type' features of colloidal aggregation in dense solutions. The model shows remarkably good agreement with experimental data and also explains why measurements on dilute solutions failed to exhibit the peak at finite q.
29.
Sciortino F, Tartaglia P: Structure factor scaling during irreversible cluster-cluster aggregation. Phys Rev Lett 1995, 74:282-285. A mean field model is proposed to describe the evolution of the structure factor during irreversible DLCA The predictions of the model are compared with experimental data [15] and dynamic simulations, and the agreement is excellent.
30. ••
Sciortino F, Belloni A, Tartaglia P: Irreversible diffusion-limited cluster aggregation: the behavior of the scattered intensity. Phys Rev £ 1995, 52:4068-4079. This paper discusses in great detail the model presented in [301. based on two equations which control the growth of the average cluster mass and the time dependence of the cluster number concentration. It provides an exhaustive explanation of the vast majority of the features of DLCA in dense solutions, and is in extremely good agreement with experimental data.
31.
32.
Zhu PW, White JW: Small angle X-ray scattering of an ordering aggregation. J Chem Phys 1996, 104:9169-9173.
33.
Dietler G, Aubert C, Cannell OS, Wiltzius P: Gelation of colloidal silica. Phys Rev Lett 1986, 57:3117-3120.
34.
Ferri F, Frisken BJ, Cannell OS: Structure of silica gels. Phys Rev Lett 1991, 67:3626-3629.
35.
Cipelletti L, Carpineti M, Giglio M: Fractal morphology, spatial order, and pore structure in microporous membrane filters. Langmuir 1996, 12:6446-6451-
36.
MacMahon P, Snook I: Small angle scattering from porous mass fractal solids. J Chem Phys 1996, 105:2223-2227.
37.
Sperry PR, Hopfenberg HB, Thomas NL: Flocculation of latex by water-soluble polymers: experimental confirmation of a nonbridging nonadsorptive, volume-restriction mechanism. J Colloid Interlace Sci 1981, 82:62-76.
15.
Carpineti M, Giglio M: Spinodal·type dynamics in fractal aggregation of colloidal clusters. Phys Rev Lett 1992, 68:3327-3330.
16.
Gunton JD, San Miguel M, Sanhi PS: In Phase Transition and Critical Phenomena. Edited by Domb C, Lebowitz JL New York: Academic Press; 1983.
17.
Degiorgio V, Banfi GP, Righini G, Rennie AR: Small-angle neutron scattering study of semiconductor microcrystallites in optical glasses. Appl Phys Lett 1990 57:2879-2881.
38.
De Hek H, Vrij A: Interactions in mixtures of colloidal silica spheres and polystyrene molecules in cyclohexane. J Colloid Interlace Sci 1981, 84:409-422.
18.
Banfi GP, Degiorgio V, Rennie AR, Barker JG: Small-angle neutron scattering study of crystal growth in semiconductordoped glasses. Phys Rev Lett 1992, 69:3401-3404.
39.
Gast AP, Hall CK, Russel WB: Polymer-induced phase separations in nonaqueous colloidal suspensions. J Colloid Interlace Sci 1983,96:251-267.
19.
Schatzel K, Ackerson BJ: Observation of density fluctuations during crystallization. Phys Rev Lett 1992, 68:337-340.
40.
20.
Bibelle J, Mason TG, Gang H, Weitz DA: Kinetically-induced ordering in gelation of emulsions. Phys Rev Lett 1992, 69:981-984.
Oosawa F, Asakura S: On interaction between two bodies immersed in a solution of macromolecules. J Chem Phys 1954, 22:1255-1256.
41.
Vrij A: Polymer at interfaces and the interactions in colloidal dispersions. Pure Appl Chem 1976,48:471-483.
42.
Ye X, NarayananXY, Tong P, Huang JS: Neutron scattering study of depletion interactions in a colloid-polymer mixture. Phys Rev Lett 1996, 76:4640-4643.
21.
Robinson OJ, Earnshaw JC: long range order in two dimensional fractal aggregation. Phys Rev Lett 1993, 71:715-718.
250
Scattering and surface forces
The structure factor of a colloid-polymer mixture, as determined at various polymer concentrations, is interpreted in terms of the interparticle potential proposed in [41).
43.
44. ••
Lekkerkerker HNW, Poon WCK, Pusey PN, Stroobants A, Warren P8: Phase behaviour of colloid-polymer mixtures. £urophys Lett 1992, 20:559-564. Poon WCK, Pusey PN, Lekkerkerker HNW: Colloids in suspense.
Phys World 1996, 9(4):27-32.
A comprehensive introduction to colloidal systems is given by reviewing the most relevant experimental results recently obtained. Analogies with the states of atomic systems are highlighted.
45.
lIett SM, Orrock A, Poon WCK, Pusey PN: Phase behaviour of a model colloid-polymer mixture. Phys Rev £ 1995, 51:1344-1352.
Verhaegh NAM, van Duijneveldt JS, Dhont JKG, Lekkerkerker HNW: Fluid-fluid phase separation in colloidpolymer mixtures studied with small angle light scattering and light microscopy. Physica A 1996, 230:409-436. The first experimental evidence of a fluid·fluid phase separation in a colloid-polymer mixture is reported. The process is thoroughly investigated by means of SALS and microscopy.
46.
47.
Poon WCK, Selfe JS, Robertson MS, IIett SM, Pirie AD, Pusey PN: An experimental study of a model colloid-polymer mixture.
J Phys 111993, 3:1075-1086. 48.
Pusey PN, Pirie AD, Poon WCK: Dynamics of colloid-polymer mixtures. Physica A 1993, 201 :322-331.
49.
Poon WCK, Pirie AD, Pusey PN: Gelation in colloid-polymer mixtures. Faraday Discuss 1995, 101 :65-76.
Poon WCK, Pirie AD, Haw MD, Pusey PN: Non-equilibrium behaviour of colloid-polymer mixtures. Physica A 1997, 235:110-119. A variety of non·equilibrium processes is observed during phase separation of colloid-polymer mixtures,including nucleation-like behaviour, spinodal-like behaviour, and transient gelation. In the three cases, rings with a characteristic time evolution are seen in SALS profiles. The presence of a hidden gas-liquid line limiting the non-equilibrium behaviour region is discussed.
50.
51.
Haw MD, Sievwright M, Poon WCK, Pusey PN: Cluster-cluster gelation with finite bond energy. Adv Colloid Interface Sci 1995, 62:1-16. Numerical simulations of aggregation in the presence of finite bond energy are performed. The structure of the aggregating system is seen to depend on the interparticle interaction potential well depth, and compact clusters failing to form a colloidal gel are found in the case of reversible aggregation.
Verhaegh NAM, Asnaghi D, Lekkerkerker HNW, Giglio M, Cipelletti L: Transient gelation by spinodal decomposition in colloid-polymer mixtures. Physica A 1997, in press. Formation of nonfractal transient gels in phase-separating colloid-polymer mixturesis reported. Scattered intensity distributions exhibiting a peak at very small angle and a Porod decay (I[ql- q-4) at larger angle are shown.
52.
53.
PiazzaR, Di Pietro G: Phase separation and gel-like structures in mixtures of colloids and surfactant £urophys Lett 1994, 28:445-450.
Small angle light scattering studies concerning aggregation processes Daniela Asnaghi, Marina Carpineti, Marzio Giglio* and Alberto Vailati Substantial work, both experimental and theoretical, has been performed on aggregation processes in dense solutions. Aggregation driven by depletion interactions has recently been studied in a variety of systems. The phase diagrams that have been obtained are extremely complex, and they include aggregation and gelation phenomena.
Addresses Dipartimento di Fisica e Istituto Nazionale per la Fisica della Materia (INFM), via Celoria 16, 20133 Milano, Italy "e-mail: [email protected] Current Opinion in Colloid & Interface Science 1997, 2:246-250 Electronic identifier: 1359-0294-002-00246 © Current Chemistry Ltd ISSN 1359-0294 Abbreviations OLCA diffusion-limited cluster aggregation OWS diffusing-wave spectroscopy RLCA reaction-limited cluster aggregation SALS small angle light scattering SANS small angle neutron scattering SAXS small angle X-ray scattering
more biological interest will also be discussed. Results in the two year period under scrutiny here will then be presented. Recent results concerning dense solutions will be described in the second section. These studies have unveiled interesting analogies between colloids and other more general thermodynamic systems. We will cover in the third section aggregation phenomena and low angle scattering from depletion interaction colloidal systems. We will not cover work done with diffusing-wave spectroscopy (DWS), in spite of the fact that it deals with scattering of light, often collected at very small angles. DWS for its own nature yields information on small length scales, and this is at variance with the technique we deal with here (DWS is discussed by Grier (pp 264-270) in another review in this issue of Current Opinion in Colloid & Interface Science). We will take, however, the liberty of quoting low angle work done at other wavelengths (neutron or X-ray) if the matter under discussion is very close to that highlighted here in conjunction with SALS.
Introduction
Fractals and universality in dilute solutions
Aggregation phenomena are fairly ubiquitous and they appear in a vast number of physico-chemical systems. Although they have been a classical topic in physical chemistry for many years, interest in them picked up momentum in the late eighties when it was realized that colloidal aggregation leads to fractal structures and that aggregation processes are endowed with universal properties.
The early works were in connection with irreversible colloidal aggregation in dilute solutions. A very important achievement was the discovery that there are two limiting universal classes of aggregation kinetics [1], namely the diffusion-limited cluster aggregation (DLCA) [2] and the reaction-limited cluster aggregation (RLCA) [3]. The former occurs when the growth is fast and controlled by pure diffusive motion, and the latter when the growth is hindered by a (weak) repulsion which makes necessary many close encounters before a bond is formed. Both regimes are characterized by universal properties: aggregates' fractal dimension, reaction kinetics, and cluster size distribution. Ingenious analysis via scattering from individual clusters has been devised [4], and cluster size distributions for the different universal aggregation routes have been determined. The same technique has reappeared in recent work [5]. The intensity distribution as a function of the scattering wave vector q yields quantitative estimates of the fractal dimension df through the asymptotic, large q power law dependence I(q) cc q-df [6]. Also, the average gyration radius and mass can be obtained as a function of time, and this is valuable for the analysis of the reaction kinetics [7.].
The formation of large structures very naturally calls for techniques like light scattering, small angle static light scattering (SALS) in particular. SALS systems are miniature replicas of the small angle neutron and X-ray scattering systems (SANS and SAXS, respectively), the scattered radiation .being collected by multi-element sensors covering a narrow range of angles near the forward direction. This detection is often done directly, without image-forming optics, and a large fraction of the scattered signal is actually collected by the sensor. The added difficulty here is that diffraction and interference effects can severely affect the performance. Diffraction spilling from the main beam can add stray light at very small angles, and random interference of the scattered light (speckles) can introduce large statistical errors. We will cover very briefly in the first section the origins of the recent work on colloidal systems undergoing aggregation in dilute solutions, and a few studies of
Light scattering studies have qualitatively shown that protein solutions can also lead to aggregation processes [8,9]_ A fairly recent and thorough study that uses, among other techniques, light scattering and SANS has
Small angle light scattering studies Asnaghi et al.
shown quite convincingly that heat-induced denaturation of proteins in solution may generate clusters that fit very well into the RLCA scheme [10). A further study [11), with SANS alone, also investigated the intraparticle structure of clusters that originated in a heat set aggregation of proteins at a different pH . Once the clusters are diluted, they show a typical DLCA structure. At denser solutions, however, anticorrelation effects of the type described in the next paragraph are also reponed. Some evidence of salt-induced clustering in protein solutions (lysozirne) has also been presented [12).
Figure 1 70
60
0
~
•A
50
~
'c ::J
.ci .... Recently, reversible aggregation, which occurs when the typical bonding energy is of the order of the particle thermal energy, has also been investigated. Using SALS, it has been shown that aggregation leads co clusters that are nonfracral [13) and that the fragmentation might be described in terms of the same universal laws which control the aggregation [14-].
247
~
-.!:!
y
<> lIE
40
•
0
A
30
v
1=398 s t=579 s 1=698 s 1=894 s t=1036 s t=1122 s t=1213s t=1302 s t=1572 s 1..1643 s
20
10
O~-'----I._"--=-:;:....J=--="""'=-...Iii.J'--""""---""--'--
Aggregation in dense solutions Studies performed in dense colloidal solutions have evidenced intriguing similarities between irreversible processes in quite different systems. Whereas in dilute solutions, the scattered intensity as a function of angle decays monotonically, in dense solutions a peak at a finite wave vector was reported for reactions occurring in the OLGA mode [lSI (see Fig. O. Such a feature is typically observed during spinod al decomposition, the phase separation process of thermodynamically unstable systems [16]. Another intriguing fact is that the scattered intensity distributions I(q) at various times seem co possess the same scaling properties already found for the spinodal decomposition case: I(q/qm) =' q m-JF(q/qm), where qm is the peak position, and F(q/qm) is a time-independent scaling function. For ordinary spinodal decomposition d ~ 3, whereas in the present case d =' dr, where dr is the fractal dimension. A plot of the scaled intensity distributions is shown in Figure 2. The appearance of a peak in the scattered pattern was almost simultaneously reponed for a variety of systems, including semiconductor crystallites grown in glass matrices [17,18), crystallization of colloidal spheres (19), liquid emulsion aggregates [20), and colloidal aggregation in two dimensions [21). Recently, a peak in the scattered distribution has also been observed during the formation of gels, when gelation competes with two-phase separation processes [22,23). During spinodal decomposition, a bicontinuous structure of domains of alternating density is created, and it is this quasi-sinusoidal modulation that generates a peak at a finite wave vector in the scatte red pattern. Quite generally, the fact that I(q =0) =0 implies that the integral of the net density correlation function vanishes. For aggreg ation processes, this means that the clusters grow by feeding on a nearby depletion region over which mass is conserved. Computer simulations have reproduced these features both in two [24-26) and three dimensions [27-).
o
5000
10000
15000
20000
25000
30000
q (em") o 1997 Current Op inion in Conoid & Inlerlace Science
Intensity distributions as collected at various times during a OLGA process in a highly concentrated sample. Whereas in dilute solutions the intensity as a function of angle decays monotonically, in this case a peak at a finite q. moving in time towards zero, can be observed.
In the 3D case [27-], reaction-limited aggregation has also been simulated, and a finite-q peak in I(q) is found, at variance with preliminary experimental results [28]. Some works have tried to explain the presence of a peak in the scattered intensity and how it evolves in time. A phenomenological model based on mass conservation arguments [29-) has been proposed, and an analytical expression for the intensity distribution in good agreement with the experimental data has been found. Furthermore, a mean field model has been proposed, based on two equations which control the growth of the average cluster mass and the time-dependence of the cluster number concentration [30--,31--) . The predictions of this model are compared with experimental dat a [IS) and Brownian dynamics simulations and the agreement is excellent. According to this theory, the scaling in the later stages of aggregation is only apparent, as the rate of growth of the depletion region is slightly different from that of the growing cluster and the system is not describable in terms of a unique length. A correlation function (32), which predicts a scattering peak at finite wave vector, has been proposed and SAXS data are fitted with this new model. Finally, we point out that in dense solutions the aggregation eventually stops when the intercluster distance equals the average cluster diameter. This means that the aggregates form an interconnected network (colloidal gel) [33] which fills all the available volume.
248
Scattering and surface forces
that the interparticle potential proposed in [41] describes well the structure factor measured at various polymer concentrations by SANS.
Figure 2
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1997 Current Opinion in Colloid & Interlace Science
Pial of Ihe scaled curves I(q/qmlqmd f, where qm is Ihe peak position, and d t is the fractal dimension. The data refer 10 the later stages of the process shown in Figure 1. The intensity distributions have scaling properties similar to those found for the spinodal decomposition case.
It has been ascertained, with SALS and SAXS, that some porous media [34-36] exhibit features which are strongly similar to those of colloidal gels. In particular, they can be described as a matrix of closely packed fractal aggregates which retain their individual structure.
Aggregation and depletion interactions Recently, the attention of the colloid physics community has been gained by very interesting systems, namely colloid-polymer mixtures; there has been pioneering experimental work in the area that dates back to the early eighties [37-39]. The presence of the polymer gives rise to attractive 'depletion-induced' forces [40,41]. The origin of these forces is simple to understand qualitatively. Let us call the radius of the colloid R, and the polymer gyration radius rG. If the colloidal particles come to a distance d (skin-skin distance) smaller than rG, then there is a region between the particles into which the polymers cannot get. As a consequence, the osmotic pressure due to Brownian collisions on the outer regions is not balanced, and a net attractive force results. The system is extremely good to work with because the interparticle potential can be tailored at will by adjusting the range of the -attractive forces (controlled by rG), and the depth of the potential well (controlled by the polymer concentration and by the ratio rdR). In a recent paper [42°], it was shown
The phase diagram of colloid-polymer mixtures is exrrernely complex and can include virtually all the features of a monatomic fluid [43,44°°,45]. For small values of the ratio rcfR, one obtains fluid-crystal coexistence, whereas for larger values the phase diagram exhibits a fluid-fluid (vapour-liquid) boundary which can display a critical point and a rricritical point. A quantitative study of fluid-fluid phase separation kinetics with SALS has been recently presented [46°]. A peak at a finite angle, collapsing to q = 0 has been observed and discussed in terms of spinodal decomposition features. In addition to phase transitions of the type encountered with simple fluids, however, depletion interaction systems also display reversible aggregation and gelation. SALS again proves to be of great help in these studies. Experimental investigations [47-49,50°°] on a system with a small ratio, namely rdR =0.08, have shown chat, indeed, a ring appears and rapidly collapses to a stationary ring at very small angle, in a way very similar to that observed in aggregation phenomena in dense colloidal solutions [15]. At this point, the system attains a gel state which exhibits fractal morphology. This gel is transient and ultimately undergoes a liquid-solid separation with the associated onset of macroscopic sedimentation and disappearance of the scattering ring. The transition from a fractal gel state to a restructured configuration is very likely due to the fact that depletion forces naturally favour dense packing [51°]. Experimental studies on systems with different rcfR ratios have shown that gels with various morphologies can be obtained. Nonfractal gels have been observed in colloid-polymer mixtures with a large ratio (rcfR = 0.25) [52°], whereas a fractal gel, although characterized by a monotonously decaying scattered intensity distribution, has been found in mixtures with rcfR=0.03 [53].
Conclusions In the period considered in the present survey, little work has been done in the classical area of aggregation in dilute solutions. l\luch more interest has been attracted by aggregation in dense solutions, because of the added complexity due to intercluster interactions. Particularly interesting is the area of depletion-induced forces. These systems show a remarkable complexity of phase diagrams, and their experimental exploration is eased by the fact that interactions can be easily controlled. Aggregation and gelation processes show novel features, and it has emerged that studies in this area may lead to a better understanding of fundamental aggregation processes in biological samples, as the interactions involved are similar to those in depletion interaction systems.
Small angle light scattering studies Asnaghi et a/.
References and recommended reading
22.
Asnaghi 0, Giglio M, Bossi A. Righetti PG: large-scale microsegregation in polyacrylamide gels (spinodal gels), J Chem Phys 1995. 102:9736-9742.
23.
Tromp RH, Rennie AR, Jones RAL: Kinetics of the simultaneous phase separation and gelation in solutions of dextran and gelatin. Macromolecules 1995, 28:4129-4138.
24.
Sintes T, Toral R, Chackrabarti A: Dynamical scaling of fractal aggregates in dense colloidal solutions. Phys Rev £ 1994, 50:3330-3333.
25.
Earnshaw JC, Robinson OJ: Inter-cluster scaling in twodimensional colloidal aggregation. Physica A 1995,214:23-51.
26.
Earnshaw JC, Harrison MBJ, Robinson OJ: local order in two-dimensional colloidal aggregation. Phys Rev £ 1996, 53:6155-6163.
Papers of particular interest, published within the annual period of review, have been highlighted as: • ••
of special interest of outstanding interest
1.
Family F, Landau DP (Eds): Kinetics 01Aggregation and Gelation. Amsterdam: North-Holland; 1984.
2.
Lin MY, Lindsay HM, Weitz DA, Klein R, Ball RC, Meakin P: Universal diffusion-limited colloid aggregation_ J Phys Condens Matt 1990, 2:3093-3113.
3.
Lin MY, Lindsay HM, Weitz DA, Ball RC, Klein R, Meakin P: Universal reaction-limited colloid aggregation. Phys Rev A 1990,41:2005-2011.
4.
Broide ML, Cohen RJ: Experimental evidence of dynamic scaling in colloidal aggregation. Phys Rev Lett 1990, 64:2026-2029.
5.
Fernandez-Barbero A, Schmitt A, Cabrerizo-VilchezM, MartinezGarcia R: Cluster-size distribution in colloidal aggregation monitored by single-cluster light scattering. Physica A 1996, 230:53-74.
6.
Sinha SK: Scattering from fractal structures. Physica 0 1989, 38:310-314.
Asnaghi 0, Carpineti M, Giglio M, Vailati'A: light scattering studies of aggregation phenomena. Physica A 1995, 213:148-158. A review of the most interesting results in the field of colloidal aggregation is presented.
7.
8.
Patkowski A, Gulari E, Chu B: long range tRNA-tRNA electrostatic interactions in salt-free and low-salt tRNA solutions. J Chem Phys 1980, 73:4178-4184.
9.
Giordano R, Maisano G, Mallamace F, Micali N, Wanderlingh F: Structural properties of macromolecular solutions. J Chem Phys 1981, 75:4770-4775.
10.
Gimel J·C, Durand 0, Nicolai T: Structure and distribution of aggregates formed after heat-induced denaturation of globular proteins. Macromolecules 1994, 27:583-589.
11.
Renard 0, Axelos MAV, Boue F,Lefebvre J: 'Ordered' structure in solutions and gels of a globular protein as studied by small angle neutron scattering. Biopolymers 1996, 39:149-159.
12.
Minezaki Y, Niimura N, Ataka M, Katsura T: Small angle neutron scattering from lysozyme solutions in unsaturated and supersaturated states (SANS from lysozyme solutions). Biophys Chem 1996, 58:355-363.
13.
Broide ML, Garrabos Y, Beysens 0: Nonfractal colloidal aggregation. Phys Rev £ 1993, 47:3768-3771.
NarayananT, Petit J-M, Broide ML, Beysens 0: Desorptioninduced fragmentation of silica aggregates. Phys Rev £ 1995, 51 :4580-4584. Small angle light scattering is used for studying the dynamics of fragmenta· tion in reversible colloidal aggregation phenomena
14.
249
Gonzalez AE, Ramirez-Santiago G: Spatial ordering and structure factor scaling in the simulations of colloid aggregation. Phys Rev Lett 1995, 74:1238-1241. 3D simulations of colloid aggregation are presented. The appearance of a peak in the I(q) is explained in terms of density modulations. A finite-q peak from reaction-limited aggregation is also found.
27.
28.
Carpineti M, Giglio M: Transition from semiorder to disorder in the aggregation of dense colloidal solutions. Phys Rev Lett 1992, 70:3828-3831.
Carpineti M, Giglio M, Degiorgio V: Mass conservation and anticorrelation effects in the colloidal aggregation of dense solutions. Phys Rev £ 1995, 51 :590-596. A phenomenological model is presented to understand the 'spinodal type' features of colloidal aggregation in dense solutions. The model shows remarkably good agreement with experimental data and also explains why measurements on dilute solutions failed to exhibit the peak at finite q.
29.
Sciortino F, Tartaglia P: Structure factor scaling during irreversible cluster-cluster aggregation. Phys Rev Lett 1995, 74:282-285. A mean field model is proposed to describe the evolution of the structure factor during irreversible DLCA The predictions of the model are compared with experimental data [15] and dynamic simulations, and the agreement is excellent.
30. ••
Sciortino F, Belloni A, Tartaglia P: Irreversible diffusion-limited cluster aggregation: the behavior of the scattered intensity. Phys Rev £ 1995, 52:4068-4079. This paper discusses in great detail the model presented in [301. based on two equations which control the growth of the average cluster mass and the time dependence of the cluster number concentration. It provides an exhaustive explanation of the vast majority of the features of DLCA in dense solutions, and is in extremely good agreement with experimental data.
31.
32.
Zhu PW, White JW: Small angle X-ray scattering of an ordering aggregation. J Chem Phys 1996, 104:9169-9173.
33.
Dietler G, Aubert C, Cannell OS, Wiltzius P: Gelation of colloidal silica. Phys Rev Lett 1986, 57:3117-3120.
34.
Ferri F, Frisken BJ, Cannell OS: Structure of silica gels. Phys Rev Lett 1991, 67:3626-3629.
35.
Cipelletti L, Carpineti M, Giglio M: Fractal morphology, spatial order, and pore structure in microporous membrane filters. Langmuir 1996, 12:6446-6451-
36.
MacMahon P, Snook I: Small angle scattering from porous mass fractal solids. J Chem Phys 1996, 105:2223-2227.
37.
Sperry PR, Hopfenberg HB, Thomas NL: Flocculation of latex by water-soluble polymers: experimental confirmation of a nonbridging nonadsorptive, volume-restriction mechanism. J Colloid Interlace Sci 1981, 82:62-76.
15.
Carpineti M, Giglio M: Spinodal·type dynamics in fractal aggregation of colloidal clusters. Phys Rev Lett 1992, 68:3327-3330.
16.
Gunton JD, San Miguel M, Sanhi PS: In Phase Transition and Critical Phenomena. Edited by Domb C, Lebowitz JL New York: Academic Press; 1983.
17.
Degiorgio V, Banfi GP, Righini G, Rennie AR: Small-angle neutron scattering study of semiconductor microcrystallites in optical glasses. Appl Phys Lett 1990 57:2879-2881.
38.
De Hek H, Vrij A: Interactions in mixtures of colloidal silica spheres and polystyrene molecules in cyclohexane. J Colloid Interlace Sci 1981, 84:409-422.
18.
Banfi GP, Degiorgio V, Rennie AR, Barker JG: Small-angle neutron scattering study of crystal growth in semiconductordoped glasses. Phys Rev Lett 1992, 69:3401-3404.
39.
Gast AP, Hall CK, Russel WB: Polymer-induced phase separations in nonaqueous colloidal suspensions. J Colloid Interlace Sci 1983,96:251-267.
19.
Schatzel K, Ackerson BJ: Observation of density fluctuations during crystallization. Phys Rev Lett 1992, 68:337-340.
40.
20.
Bibelle J, Mason TG, Gang H, Weitz DA: Kinetically-induced ordering in gelation of emulsions. Phys Rev Lett 1992, 69:981-984.
Oosawa F, Asakura S: On interaction between two bodies immersed in a solution of macromolecules. J Chem Phys 1954, 22:1255-1256.
41.
Vrij A: Polymer at interfaces and the interactions in colloidal dispersions. Pure Appl Chem 1976,48:471-483.
42.
Ye X, NarayananXY, Tong P, Huang JS: Neutron scattering study of depletion interactions in a colloid-polymer mixture. Phys Rev Lett 1996, 76:4640-4643.
21.
Robinson OJ, Earnshaw JC: long range order in two dimensional fractal aggregation. Phys Rev Lett 1993, 71:715-718.
250
Scattering and surface forces
The structure factor of a colloid-polymer mixture, as determined at various polymer concentrations, is interpreted in terms of the interparticle potential proposed in [41).
43.
44. ••
Lekkerkerker HNW, Poon WCK, Pusey PN, Stroobants A, Warren P8: Phase behaviour of colloid-polymer mixtures. £urophys Lett 1992, 20:559-564. Poon WCK, Pusey PN, Lekkerkerker HNW: Colloids in suspense.
Phys World 1996, 9(4):27-32.
A comprehensive introduction to colloidal systems is given by reviewing the most relevant experimental results recently obtained. Analogies with the states of atomic systems are highlighted.
45.
lIett SM, Orrock A, Poon WCK, Pusey PN: Phase behaviour of a model colloid-polymer mixture. Phys Rev £ 1995, 51:1344-1352.
Verhaegh NAM, van Duijneveldt JS, Dhont JKG, Lekkerkerker HNW: Fluid-fluid phase separation in colloidpolymer mixtures studied with small angle light scattering and light microscopy. Physica A 1996, 230:409-436. The first experimental evidence of a fluid·fluid phase separation in a colloid-polymer mixture is reported. The process is thoroughly investigated by means of SALS and microscopy.
46.
47.
Poon WCK, Selfe JS, Robertson MS, IIett SM, Pirie AD, Pusey PN: An experimental study of a model colloid-polymer mixture.
J Phys 111993, 3:1075-1086. 48.
Pusey PN, Pirie AD, Poon WCK: Dynamics of colloid-polymer mixtures. Physica A 1993, 201 :322-331.
49.
Poon WCK, Pirie AD, Pusey PN: Gelation in colloid-polymer mixtures. Faraday Discuss 1995, 101 :65-76.
Poon WCK, Pirie AD, Haw MD, Pusey PN: Non-equilibrium behaviour of colloid-polymer mixtures. Physica A 1997, 235:110-119. A variety of non·equilibrium processes is observed during phase separation of colloid-polymer mixtures,including nucleation-like behaviour, spinodal-like behaviour, and transient gelation. In the three cases, rings with a characteristic time evolution are seen in SALS profiles. The presence of a hidden gas-liquid line limiting the non-equilibrium behaviour region is discussed.
50.
51.
Haw MD, Sievwright M, Poon WCK, Pusey PN: Cluster-cluster gelation with finite bond energy. Adv Colloid Interface Sci 1995, 62:1-16. Numerical simulations of aggregation in the presence of finite bond energy are performed. The structure of the aggregating system is seen to depend on the interparticle interaction potential well depth, and compact clusters failing to form a colloidal gel are found in the case of reversible aggregation.
Verhaegh NAM, Asnaghi D, Lekkerkerker HNW, Giglio M, Cipelletti L: Transient gelation by spinodal decomposition in colloid-polymer mixtures. Physica A 1997, in press. Formation of nonfractal transient gels in phase-separating colloid-polymer mixturesis reported. Scattered intensity distributions exhibiting a peak at very small angle and a Porod decay (I[ql- q-4) at larger angle are shown.
52.
53.
PiazzaR, Di Pietro G: Phase separation and gel-like structures in mixtures of colloids and surfactant £urophys Lett 1994, 28:445-450.