Model of Polydisperse Wormlike Stars and Its Application to Dyestuff

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Model of Polydisperse Wormlike Stars and Its Application to Dyestuff Aggregates Armin Katzenstein† and Klaus Huber* Universita¨ t-GH Paderborn, Fachbereich 13 Chemie und Chemietechnik, Warburger Str. 100, D-33098 Paderborn, Germany Received February 11, 2002. In Final Form: June 24, 2002 A description of the scattering behavior of combinatorial wormlike stars, based on Koyama’s approximation for linear wormlike chains, was extended to polydisperse star samples. The number of arms per star was constant. All arms within a single star-branched particle had the same length. Polydispersity was accounted for by merely using a Schulz-Zimm type of distribution for the overall contour length of the branched wormlike particles. Applicability of this model was tested on aggregates of anionic azo dyestuffs. The aggregation process was induced by the addition of low amounts of MgSO4 to aqueous dyestuff solutions and recorded on-line by means of time-resolved multiangle static light scattering. Each aggregation run resulted in a large number of scattering curves as a function of the aggregation time. A first interpretation of the scattering curves with the model of linear wormlike chains led to acceptable parametrization with an averaged persistence length of 180 nm. However, application of the combinatorial wormlike star model indicated a likelihood for slight branching. Thereby, a reliability limit of the linear wormlike chain model was revealed if branching of the growing linear aggregates sets in.

Introduction Several classes of low molecular weight compounds are able to form gels from solution, both in organic solvents and water.1,2 One of these classes includes anionic azo dyestuffs which have been produced and commercialized on an industrial scale for more than 100 years. They turned out to be notorious gelators in water. Attention has been drawn to this behavior from the very beginning3 because gelation raises problems during synthesis and refinement of the dyestuffs, and it interferes with dyestuff adsorption during the dyeing process. Interest in gelation (and aggregation) of dyestuffs persists to this very day. Further insight into the physicochemical aspects of gelation of dyestuffs helps to solve problems occurring during the production and application of dyestuffs, and comparison to other self-organizing systems of small synthetic molecules and of biopolymers4 could provide fascinating analogies. Clearly, isolated aggregates can be considered as being predecessors of the macroscopic gel. Usually, the aggregates grow to large filaments which, at some point, meet at junctions. We intend to study parameters which induce and affect the growth and branching during aggregation/gelation. In this respect, mass, size, and shape of primary dyestuff aggregates are relevant properties. In the past, a variety of methods5,6 were used to investigate these properties and their dependencies on environmental * To whom correspondence should be addressed. E-mail: huber@ chemie.uni-paderborn.de. † Present address: Sovias AG, Physikalische Chemie, WKL127.5.86, Postfach, CH-4002 Basel, CH. (1) Fuhrhop, J.-H.; Helfrich, W. Chem. Rev. 1993, 93, 1565. (2) Terech, P.; Weiss, R. G. Chem. Rev. 1997, 97, 3133. (3) Haller, R. Kolloid-Z. 1918, 22, 4. (4) Alberst, B.; Bray, D.; Lewis, J.; Raff, M.; Roberts, K.; Watson, J. D. The Molecular Biology of the Cell, 3rd ed.; Garland Publishers: New York, 1994 and references therein. (5) Burdett, B. C. In Aggregation Processes in Solution; Wyn-Jones, E., Gormally, J., Eds.; Elsevier Science: New York, 1983; p 241. (6) Edwards, D. E.; Ormerod, A.; Tiddy, G. J. T.; Jabar, A. A.; Mahendrasingham, A. In Physico-chemical principles of color chemistry; Peters, A. T., Freeman, H. S., Eds.; Advances in Color Chemistry Series Vol. 4; Blackie Academic&Professional: London, 1996; p 83.

conditions. A few correlations could be established in the meantime: Aggregate formation is accelerated by the addition of inert salts or the decrease of temperature. On the other hand, organic additives such as urea or phenol have the contrary effect. In several cases, no equilibrium state was reached and the particles gradually grew for hours or even days. Further work by Lydon et al.7 and Tiddy et al.8 showed that anionic azo dyestuffs formed lyotropic mesophases in aqueous solution. X-ray diffraction indicated the ordering of rodlike aggregates. Interestingly enough, Tiddy et al.8 suggested a shape for CI Acid Red 266 aggregates similar to the tube model of polymerized tubulin as one of the constituent proteins of the cytoskeleton. Finally, Imae et al. succeeded in visualizing a highly branched network of dyestuff filaments by means of electron microscopy in water/methanol mixed solvents.9 Like other gelators, anionic azo dyestuffs exert various specific interactions. Those interactions include H-bonds, π-π interactions, or van der Waals forces,6 resulting in a capability to self-assemble. Together with the tendency of the resulting aggregates to form three-dimensional networks, this suggests distinct analogies to chain growth and branching of macromolecules. In both cases, oriented forces during bond formation lead to complex structures with a high degree of order. They can be regarded as two extremes which limit a wide scope of variable stability reaching from chemically stable macromolecules with remarkable mechanical properties to highly reversible systems open to adjustable assemblies. It is obvious that various theoretical concepts originally developed in polymer science can be extended to describe reversible aggregates. At the same time, experimental techniques to investigate size and shape of polymers are equally suitable for the characterization of polymeric aggregates. Those techniques include dynamic light scattering (DLS) and (7) Attwood, T. K.; Lydon, J. E.; Jones, F. Liq. Cryst. 1986, 1, 499. (8) Tiddy, G. J. T.; Mateer, D. L.; Ormerod, A. P.; Harrison, W. J.; Edwards, D. J. Langmuir 1995, 11, 390. (9) Imae, T.; Gagel, L.; Tunich, Ch.; Platz, G.; Iwamoto, T.; Funayama, K. Langmuir 1998, 14, 2197.

10.1021/la020143u CCC: $22.00 © 2002 American Chemical Society Published on Web 08/09/2002

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Katzenstein and Huber

static light scattering (SLS). In fact, light scattering has become an excellent tool to investigate aggregating systems in general. Since the pioneering work on lauryl sulfate by Mysles and Princen,10 a large number of publications on amphiphiles have appeared in the literature. Also, researchers have occasionally looked at aggregation of azo dyes.11-21 It eventually turned out that the model of wormlike chains, originally developed in order to describe real polymer chains,22 is just as useful for describing elongated chainlike surfactant aggregates.23 Interpretation of scattering curves was initially performed24,25 by means of the Koyama approximation26 but has made significant progress since due to the work of Schurtenberger et al.27 and Ballauff et al.28 Whereas oriented interactions may readily explain formation of linear chains, branching, although it has to occur at some point along the route to macroscopic networks, has not been established with isolated azo dyestuff aggregates yet. The only indications on isolated branched aggregates from low molecular weight gelators known to us were published by Terech and Wade29 and Beginn et al.30 They succeeded in visualizing star-branched aggregates by means of electron microscopy from a steroid29 and a carbohydrate amphiphilic monomer30 in an organic solvent, respectively. Unfortunately, sample preparation for electron microscopy may impair the aggregation process or modify the resulting structures in an unpredictable manner. Therefore, complementary methods together with appropriate interpretation schemes for the resulting data are highly desirable in order to investigate branching during aggregation. Time-resolved SLS31,32 may offer a chance to fill this gap. Recently, this new method has been used successfully to investigate the aggregation process of anionic azo dyestuffs in an aqueous solution.33 Two dyestuffs were applied in a binary mixture. The aggregation process was induced by the addition of small amounts of MgSO4 and proceeded over a time interval of some 10 min. During

this time, chains of dyestuff molecules grew in length from 300 up to 2500 nm. Having available a system for time-resolved SLS as suitable as the above-mentioned dyestuff mixture, we will now extend experimental evidence for the aggregation process recently presented.33 At the same time, we will focus on a potential route to investigate and detect branching during such aggregation processes. As a starting point, an extension of the Koyama approximation for particle scattering factors of wormlike chains will be used. This extension includes regularly branched worms, denoted as combinatorial stars.34 In these combinatorial stars, any two arms of a star are freely jointed at the center, behaving as a broken wormlike chain. In the second section, we therefore briefly introduce the model of broken wormlike stars,34 which we consider appropriate to interpret experimental scattering curves of branched aggregates. In the third part, we describe the materials used and the SLS experiments performed and summarize extraction of model-independent molecular parameters from scattering data33 and interpretation of the data by means of the Kratky-Porod linear wormlike chain model. This way of interpretation was developed in the preceding paper33 and has the advantage of avoiding the drawback of the Koyama approximation.27,28 The next section presents a detailed estimation of the experimental uncertainty of the results. Based on this knowledge, reinterpretation of some of the data in terms of the model of combinatorial wormlike stars shall be given in the fifth section. It is our intention to reveal the effect of branching on the scattering behavior of wormlike chains, thereby adding insight into the impact of one further structural feature. It is comparable to the investigation by Pedersen and Schurtenberger,35 who investigated the influence of the excluded volume on the scattering behavior of wormlike chains.

(10) Mysles, K. J.; Princen, L. H. J. Phys. Chem. 1958, 63, 1696. (11) Alexander, P.; Stacy, K. A. Proc. R. Soc. London 1952, A212, 274. (12) Frank, H. P. J. Colloid Sci. 1957, 12, 480. (13) Sivarajan, S. R. J. Indian Inst. Sci. 1952, 34, 75. (14) Sivarajan, S. R. J. Indian Inst. Sci. 1954, 36, 282. (15) Banderet, A.; Meyer, P. Bull. Soc. Chim. France 1953, 53, 9. (16) Datyner, A.; Flowers, A. G.; Pailthorpe, M. T. J. Colloid Interface Sci. 1980, 74, 71. (17) Reeves, R. L.; Maggio, M. S.; Harkaway, S. A. J. Phys. Chem. 1979, 83, 2359. (18) Datyner, A.; Pailthorpe, M. T. J. Colloid Interface Sci. 1980, 76, 557. (19) Datyner, A.; Pailthorpe, M. T. Dyes Pigm. 1987, 8, 253. (20) Chavan, R. B.; Rao, J. V. J. Soc. Dyers Colour. 1983, 99, 126. (21) Neumann, B.; Huber, K.; Pollmann, P. Phys. Chem. Chem. Phys. 2000, 2, 3687. (22) Kratky, O.; Porod, G. Recl. Trav. Chim. 1949, 68, 1106. (23) Magid, L. J. J. Phys. Chem. B 1998, 102, 4064 and references therein. (24) Denkinger, P.; Kunz, M.; Burchard, W. Colloid Polym. Sci. 1990, 268, 513. (25) von Berlepsch, H.; Dautzenberg, H.; Rother, G.; Ja¨ger, J. Langmuir 1996, 12, 3613. (26) Koyama, R. J. Phys. Soc. Jpn. 1973, 34, 1029. (27) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E 1997, 56, 5772. (28) Po¨tschke, D.; Hickl, P.; Ballauff, M.; Astrand, P.-O.; Pedersen, J. S. Macromol. Theory Simul. 2000, 9, 345. (29) Terech, P.; Wade, R. H. J. Colloid Interface Sci. 1988, 125, 542. (30) Beginn, U.; Keinath, S.; Mo¨ller, M. Macromol. Chem. Phys. 1998, 199, 2379. (31) Becker, A.; Schmidt, M. Makromol. Chem., Macromol. Symp. 1991, 50, 249. (32) Egelhaaf, S. U.; Schurtenberger, P. Rev. Sci. Instrum. 1996, 67, 540. (33) Escudero Ingle´s, S.; Katzenstein, A.; Schlenker, W.; Huber, K. Langmuir 2000, 16, 3010.

Materials. Except for the counterion of the yellow dyestuff, the same two dyestuff components were used as in the preceding paper.33 The molar mass of the sodium salt of the red dyestuff (RD) is 1061 g/mol. The ammonium ion of the yellow dyestuff (YD) was HN(C2H4-OH)3+, increasing the molar mass of the YD to 632.7 g/mol compared to 505 g/mol for the Na salt of YD. The ammonium salt of the yellow dyestuff (AmYD) directly resulted from industrial production. Use of this salt instead of the sodium salt avoids necessity of complete ion exchange of the ammonium ions by Na ions. Originally, the new experiments were designed to look for differences between the aggregation process with NaYD and with AmYD. However, the new experiments on AmYD/RD mixtures rather proved the robustness of the underlying aggregation process because the results with AmYD basically led to the same behavior as found earlier33 with the NaYD. They also underlined the importance of the Mg ions in the initiation of the process. Chemical structures for both dyestuffs are shown in Figure 1. Purification of the dyestuff anions was achieved by dialysis of aqueous solutions of the raw material. Dialysis of the dyestuff components was carried out with Spectra/Por dialysis hoses from Spectrum (USA) with a specified cutoff molecular weight of 6000-8000 g/mol. Diameter and length of the hoses were 12 and 30 cm, respectively. For each dyestuff, three hoses were filled with a concentrated solution of 4 g/L of dyestuff. After being charged with 200 mL of dyestuff solution, each hose was sealed with clamps at both ends and placed in a 10 L vessel. The vessel was completely filled with the dialysis medium. Dialysis of RD was performed against an aqueous solution of 1.2 g/L NaCl corresponding to 2.05 M NaCl. The

Experiments and Data Treatment

(34) Huber, K.; Burchard, W. Macromolecules 1989, 22, 3332. (35) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602.

Model of Polydisperse Wormlike Stars

Figure 1. Chemical structures of the dyestuffs under investigation: anion of the yellow dyestuff (YD) and anion of the red dyestuff (RD). dialyzing medium was changed twice a day. At the beginning of the fourth day, aqueous NaCl was replaced by distilled water as the dialyzing medium. Now, the dialyzing medium was changed three times a day. The endpoint of the dialyzing process was indicated by subjecting the dialyzing media to a precipitation test for Cl- ions with an Ag+ solution. In all cases, the test was negative after 3 more days of dialyzing. Dialysis of AmYD was performed in an analogous manner, except for the use of 2.05 mol/L of ClNH(C2H4-OH)3 in distilled water as the dialyzing medium during the first 3 days of the procedure. Aqueous solutions of ClNH(C2H4-OH)3 were prepared by the addition of equimolar amounts of HCl and triethanolamine to freshly distilled water. After dialysis, the water content in the hoses had increased by some 50% due to water penetrating the membrane. To recover the dyestuffs, their aqueous solutions were first concentrated by rotation evaporation of about 90% of the water at 30 °C under reduced pressure. The remaining water then was removed by freeze-drying of the concentrated dyestuff solutions. Preparation of Solutions for Light Scattering Experiments. All light scattering experiments were performed with dilute solutions of a mixture of RD and AmYD. In all cases, the solvent was a solution of MgSO4 in bidistilled water at pH ) 7. Three concentrations of the magnesium sulfate, 1.63, 2.44, and 3.3 mM, were applied. The composition of the dyestuff mixture was always 50 wt % of RD and AmYD. Like the mixing ratio, the overall concentration of the dyestuff mixture was kept constant. The actual value of the dyestuff concentration was 0.0664 g/L, close to the value applied in the preceding paper on RD/NaYD mixtures.33 Separate dyestuff solutions were prepared for each component in bidistilled water. The dyestuff concentration of both stock solutions was 0.0664 g/mL. The MgSO4 solutions were prepared by dissolving MgSO4‚7H2O, pa grade (Fluka, CH), in bidistilled water. The MgSO4 concentration and the dyestuff concentration were twice as large as the values which had to be finally achieved for the scattering experiment. Before the scattering experiments were started, 10 mL of each dyestuff solution was combined in a 20 mL graduated flask. Five milliliters of this mixed dyestuff solution was put into a dustfree cuvette, and 5 mL of the MgSO4 solution was added. To remove dust, both solutions were passed through Millex-GV filters with a pore width of 0.22 µm from Millipore (USA) while being added to the cuvette. Prior to this, the filters were conditioned with 10 mL of the same solution. Time-Resolved Static Light Scattering Experiments. Cylindrical quarz cuvettes (Helma, Germany) with a diameter of 20 mm were used as scattering cells. All scattering cells were cleaned of dust by continuously injecting freshly distilled acetone from below for 15 min. The aqueous solution of the dyestuff mixture and the aqueous solution of MgSO4 were added sequentially into the scattering cell. At all three MgSO4 contents, dyestuff aggregates with a gradually increasing size were observed. The addition of the MgSO4 solution initiated aggregation and defined the starting point of time measurement. The

Langmuir, Vol. 18, No. 18, 2002 7051 time between the addition of the MgSO4 solution and the first scattering experiment was roughly half a minute. All scattering experiments were done with a model 1800 static light scattering instrument from ALV-Laser Vertriebsgesellschaft (Germany). A krypton ion laser model 2016 Stabilite from Spectra Physics (USA), operating at a wavelength of λ ) 647.1 nm, was used as a light source. For further details of the SLS instrument, the reader is referred to the preceding paper.33 In addition, a prototype of this light scattering instrument, which differs only slightly from the one used in the present work, was described recently by Becker and Schmidt.31 SLS results were expressed in terms of the Rayleigh ratio. Calculation was based on the known Rayleigh ratio of toluene. Prior to any measurement of a dyestuff solution, the scattering intensity of toluene and of the solvent was performed subsequentially at all 18 scattering angles, θ, which covered an angular range of 32° < θ < 143°. A solution of MgSO4 in bidistilled water was used as a solvent, where the MgSO4 concentration was the same as in the corresponding dyestuff solution. As for the solutions, the solvents were passed through Millex-GV filters with a pore width of 0.22 µm into the scattering cells. The solvent measurement enabled subtraction of the background scattering from the solution. Scattering data were evaluated from the difference ∆Rθ between the Rayleigh ratio of the dyestuff solution and that of the solvent background. Data are represented as a function of the square of the scattering vector, q ) (4πn/λ) sin(θ/2), with n being the refractive index of the solvent, λ the laser wavelength in vacuo (647.1 nm), and θ the scattering angle.

P(q) ) [∆Rθ/(Kc)]/[∆R0/(Kc)]

(1)

In eq 1, K is the contrast factor and c is the concentration in grams per milliliter of the dyestuff mixture. The particle scattering factor P(q) in general corresponds to ∆Rθ/(Kc) divided by its limit at q f 0 and c f 0, ∆R0/(Kc). Just as in the preceding investigation, we had to abstain from extrapolation to c f 0 because the size of the aggregates depends on c. But the use of a dyestuff concentration as small as 0.0664 g/L may justify this approximation. Evaluation of Model-Independent Parameters. Addition of MgSO4 to a solution of mixed NaYD/RD led to large polydisperse aggregates with a wormlike shape. The size distribution was characterized with the so-called polydispersity index U,

U ) Lw/Ln - 1 ) 1/z

(2)

which adopted a value of 1. In eq 2, Lw is the weight-averaged contour length and Ln is the number-averaged contour length of the wormlike aggregates. Weight-averaged contour lengths Lw varied within 300 nm < Lw < 2500 nm.33 This clearly exceeded the range where Zimm’s or even Guinier’s approximation could still be applied to extract an apparent weight-averaged molar mass Mwapp ) ∆R0/(Kc) or a z-mean square radius of gyration Rg2. Data evaluation therefore was achieved by a series expansion of Kc/∆Rθ in powers of q2. The slightly modified system AmYD/RD under present investigation yielded similar results, suggesting the same series expansion for data evaluation.

Kc/∆Rθ ) a1 + a2q2 - a3q4

(3a)

Kc/∆Rθ ) 1/Mwapp[1 + (〈S2〉z/3!)q2 - (〈S4〉z/5! - (〈S2〉z/3!)2)q4] (3b) In eq 3b, 〈S2〉z and 〈S4〉z are the z-averaged second and fourth moment of the distribution of mass within the particles with 〈S2〉z/2 ) Rg2. The q4 term accounts for the initial curvature in Kc/∆Rθ. Evaluation of all values of Mwapp ) ∆R0/(Kc) and Rg2 according to eq 3 was based on the initial part of the scattering curve, covering a q2 range of 0 < q2 < 2.5 × 10-4 nm-2. Validity of eq 3 in the range of 300 nm < Lw < 2500 nm was tested by means of model scattering curves based on Koyama’s approximation for wormlike chains with U ) 1 in advance. Evaluation of Linear Wormlike Chain Parameters. It has been established by several authors that Koyama’s ap-

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proximation26 to scattering curves of wormlike chains works poorest if qa ∼ 1. Sufficiently accurate procedures were developed only recently by Schurtenberger et al.27 and Ballauff et al.,28 however, at the expense of easily tractable expressions. To circumvent both drawbacks, a simple and alternative method was developed in the preceding paper,33 which yielded reliable wormlike chain parameters in the regime under consideration. This method solely relies on properly evaluated model-independent size parameters and on the accessibility of a plateau value in (∆Rθ/Kc)q versus q at large q. In almost all cases, those two requirements were fulfilled. The plateau value of

q ∆Rθ/(Kc) ) πMLapp ) πMwapp/Lw

(4)

leads to the linear mass density of the wormlike aggregate MLapp and by use of the weight-averaged molar mass allows calculation of the weight-averaged contour length Lw ) Mwapp/MLapp. Once the contour length was established, the persistence length a as the second parameter could be extracted by means of the analytical expression for the mean square radius of gyration Rg2th for polydisperse wormlike chains obeying a Schulz-Zimm36 distribution of the contour length.37

Rg2th )

(z + 2)aLw (z + 1)3

- a2 +

2a3 Lw

2a4(z + 1)2 2

[ (

Lw z(z + 1)

1-

)]

(z + 1)a (z + 1)a + Lw

z

(5)

The procedure is to minimize deviation ∆ from experimental Rg2,

∆ ) Rg2 - Rg2th

(6)

by varying a in Rg2th of eq 6 at z ) 1. Thereby, the value of a leading to the smallest ∆ was taken as the proper persistence length of the chain.

The Model of Combinatorial Stars with Broken Worms In an attempt to describe scattering curves of regular star-branched chains, both from small-angle neutron scattering in solution and from Monte Carlo simulations, the Koyama approximation26 was extended to model stars with a fixed number of wormlike arms, equal in lengths.34 A practicable route to achieve this goal was to freely joint the arms at a single point. This point forms a star center from which the arms emanate without orientational correlations. Any two arms of the same star can be regarded as a broken wormlike chain. As a major result, the approximation turned out to work well, as soon as the arm length exceeded a few persistence lengths and if the excluded volume effect could be neglected.34 The latter effect does even play an important role for stars in Θ solvents if the number of arms is larger than 10 and the segment density becomes fairly large. The formula derived for the particle scattering factor is

PBWstar(q) ) f-1{X-2

∫(X - x) φ(x,q) dx + (f - 1)[∫φ(x,q) dx]2}

(7)

with

φ(x,q) ) exp[-(s2/3)x f(x)] sin(sx g(x))/sx g(x) (8) being the scattering factor of a pair of monomers. In eqs (36) Zimm, B. J. Chem. Phys. 1948, 16, 1099. (37) Oberthu¨r, R. C. Makromol. Chem. 1978, 179, 2693.

Figure 2. (a) Scattering curves as q2P versus q for combinatorial wormlike stars with overall contour length Lw ) 2957 nm, persistence length a ) 220 nm, and 6 arms at varying polydispersity indices z: z ) 1 (curve 1), z ) 5 (curve 2), and z ) ∞ (curve 3). (b) Scattering curves for combinatorial wormlike stars with a polydispersity index of z ) 1, a contour length of Lw ) 18 000 nm, and 6 arms at varying persistence length a, corresponding to a varying number of persistence lengths per star Lw/a: a ) 1339 nm, Lw/a ) 13.4 (curve 1); a ) 500 nm, Lw/a ) 36 (curve 2); a ) 180 nm, Lw/a ) 100 (curve 3); a ) 100 nm, Lw/a ) 180 (curve 4). The Lw/a value of curve 1 is identical to the one of curve 1 in (a).

7 and 8, the symbols X, x, and s denote the contour length L, the contour segment length t, and the scattering vector q normalized with the persistence length a.

X ) L/a

(9a)

x ) t/a

(9b)

s ) q/a

(9c)

The terms f(x) and g(x) appearing in eq 8 are given explicitly in the Appendix of ref 34. We expect now to apply eq 7 to branched wormlike aggregates. However, if branching does occur in wormlike aggregates, it most likely will appear randomly, which means that a variety of branching points differing in number of arms, each arm differing in length, will arise. In the present work, we will account for this merely by allowing the overall contour length Lw to be spread according to a Schulz-Zimm36 type of distribution with z ) 1. The number of arms is the same for all particles, and the arm lengths do not vary within one star. Being aware of these simplifications, denoted as regular branching, we hope to finally present an argument which still might justify applicability of this star model in retrospect. Many of the publications which deal with the scattering behavior of regular stars focus on the maximum of a plot of q2P versus q denoted as a Kratky plot. This feature was first established for monodisperse regular stars with f g 4 obeying Gaussian chain statistics38 and also occurs for monodisperse regular broken wormlike stars.34 Following this route, we first describe the impact of polydispersity and a variation in the persistence length on this maximum. In Figure 2a, three scattering curves from 6-arm stars with an overall contour length of 2957 nm and a persistence length of 220 nm are compared. Being aware of the fact (38) Burchard, W. Macromolecules 1977, 10, 919.


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Figure 4. Root of z-averaged mean square radius of gyration Rg and weight-averaged contour length Lw as a function of aggregation time for the system AmYD/RD at [MgSO4] ) 1.63 mM (b), [MgSO4] ) 2.44 mM (9), and [MgSO4] ) 3.3 mM (2). Figure 3. Bending rod plots of linear and star-branched wormlike chains according to eq 7. In all cases, the polydispersity is given by z ) 1. The persistence length of all stars is a ) 180 nm. (a) Stars with varying numbers of arms at a constant arm length of 2000 nm. From top to bottom, the number of arms increases from f ) 2 (linear chain) to f ) 6. (b) Stars with varying numbers of arms at a constant overall contour length of Lw ) 12 000 nm. With the curves from top to bottom, the number of arms increases from f ) 2 (linear chain) to f ) 6. (c) Linear wormlike chains with different persistence lengths at a constant overall contour length of Lw ) 12 000 nm. Curves from top to bottom correspond to persistence lengths a ) 180 nm, a ) 135 nm, a ) 90 nm, and a ) 45 nm, respectively.

that the aggregates under present investigation have a polydispersity index close to z ) 1, it is interesting to note that a 6-arm star with z ) 1 and 2.2 persistence lengths per arm does not show this maximum. However, it appears gradually on lowering the polydispersity by increasing z from 1 to infinity. At the same time, appearance of the maximum is influenced by chain stiffness which is varied by the number of persistence lengths per arm. Assuming a constant polydispersity index of z ) 1, it does not appear before the number of persistence lengths per arm exceeds the value of 6 (Figure 2b). As a consequence, the Kratky plot seems to lose some of its ability to represent characteristic features of the scattering behavior of stars, once those stars become too stiff and/or too polydisperse, and attention is drawn to the so-called bending rod plot,39 that is, qP versus q. This plot has proven to be extremely helpful in the representation of scattering curves from linear wormlike chains. In fact, two interesting features of polydisperse broken wormlike stars could also be revealed. The first feature reveals the effect of an increasing number f of arms equal in length in a star at constant arm length (Figure 3a) and at constant overall contour length (Figure 3b). As shown in Figure 3a, semiflexible branched chains with a fixed number of arms lead to scattering curves with a maximum and a plateau. The height of the latter decreases with increasing overall contour lengths according to π/Lw. If, on the other hand, the averaged total contour length Lw is kept constant while increasing the number of arms, all curves adopt the same plateau value, independent of the number of arms. Clearly, the height (39) Denkinger, P.; Burchard, W. J. Polym. Sci., Part B: Polym. Phys. 1991, 29, 589.

of the plateau value is merely determined by the total contour length and not by the length of any two arms within a star (Figure 3b). This allows one to determine the correct mass per unit length MLapp even if one does not know in advance whether wormlike particles are branched or not. The second feature is shown in Figure 3b,c. Here, all curves are based on particles with the same overall contour length. In Figure 3c, variation is brought about by decreasing the persistence length a of a polydisperse linear wormlike chain with Lw ) 12 000 and z ) 1. As expected, an increase in flexibility increases the maximum of the curves. Interestingly enough, the same qualitative change takes place if a branching point is introduced and the number of arms is increased (Figure 3b); to keep the overall contour length constant, the arm lengths had to be shortened at the same time. This raises the question of whether it is possible at all to distinguish between an increase in flexibility and an increase in the number of arms. Results on AmYD/RD Mixtures As already observed with the NaYD/RD system,33 addition of MgSO4 influences the growth rates and the size of mixed AmYD/RD aggregates. Again, growth of the chain length was associated with an increase in the linear mass density by more than 1 order of magnitude. Although this points to an increase of the lateral dimensions of AmYD/RD aggregates as dramatic as the one observed with NaYD/RD particles, this aspect shall not be discussed any further because lack of the dn/dc value of AmYD/RD prevents access to absolute mass data in this case. Above all, it is our intention to focus on the increase in chain dimensions and on branching if it accompanies chain growth. Figure 4a,b shows the increase of chain dimensions as a function of the aggregation time t. The chain dimensions are expressed as Rg and Lw and developed according to eqs 3 and 4, respectively. For both quantities, the more MgSO4 added, the larger the extent of growth. Chain stiffness can be represented in terms of the persistence length a, resulting in an averaged value of a ) 180 nm for AmYD/RD and for NaYD/RD. A closer look at details in Figure 5a reveals even similar small trends in a for both systems when plotted versus the contour length Lw. These trends comprise a sharp increase in a at

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Figure 6. Fractional uncertainty ∆X/X as a function of experimental Lw for the system AmYD/RD including all three [MgSO4] concentrations: (dotted line) X ) a1 ) 1/Mwapp; (dasheddotted line) X ) a2; (solid line) X ) Rg2.

uncertainty of both parameters adds up to the fractional uncertainty of Rg2 according to Figure 5. Interpretation of the global dimensions in terms of the wormlike chain model with a polydispersity given by z ) 1. (a) Persistence length a evaluated according to eqs 5 and 6 at [MgSO4] ) 1.63 mM (O), [MgSO4] ) 2.44 mM (0), and [MgSO4] ) 3.3 mM (4). The drawn line indicates an average value of a ) 180 nm. (b) Radius of gyration Rg ) (〈S2〉z/2)0.5 evaluated according to eq 3 at [MgSO4] ) 1.63 mM (O), [MgSO4] ) 2.44 mM (0), and [MgSO4] ) 3.3 mM (4). The horizontal line indicates a theoretical curve according to eq 5 with z ) 1 and a ) 180 nm.

very short chains and a gradual decrease in a toward very long chains. However, disregarding these trends for a moment, we represent Rg versus Lw according to eq 5 with a single value for the parameter a. Using the averaged persistence length of a ) 180 nm, the resulting theoretical curve is shown in Figure 5b. The theoretical curve comes to lie above the experimental data if Lw gets very large. These small deviations are a direct consequence of the gradual decrease in a toward increasing chain length. It is this gradual decrease in a that will lie at the center of a careful discussion presented in the succeeding sections. The discussion considers the potential influence of branching on wormlike chain parameters of the growing aggregates, based on an appropriate evaluation of data uncertainty. Uncertainty of the Evaluated Wormlike Chain Parameters Although variation of the persistence length with increasing Lw is very small for both systems, the present data reproduce the trend observed with NaYD/RD. Indication for a decrease of a may be further strengthened if we omit data of lower precision. For example, an overlay of data sets from different aggregation runs, for example, belonging to different MgSO4 contents, may partly conceal such trends. In addition, uncertainty of a may depend on L w. Therefore, careful discussion of the data requires a reliable estimation of the experimental uncertainty in Rg2 which, in turn, depends on the uncertainty of the intersection a1 of eq 3 with the ordinate and on the uncertainty of the coefficient a2 of the q2 term. The regression procedure with eq 3 is based on the method of least squares, and aside from a1 and a2, it yields the corresponding standard deviations ∆a1 and ∆a2. Dividing by the respective fit parameters transforms both standard deviations into the corresponding fractional uncertainties ∆a1/a1 and ∆a2/a2. Finally, Rg2 corresponds to the quotient of the two fit parameters a1 and a2, and the fractional

∆a1/a1 + ∆a2/a2 ) ∆Rg2/Rg2

(10)

Although scattering signals increase with increasing particle size, the experimental uncertainties of 1/Mwapp and Rg2 do not follow a unique trend. Usually, the larger the scattering signal, the greater the accuracy of the initial slope of the scattering curves. The same can be expected for the intersection only if we start from a very low scattering signal. If the particle mass exceeds a certain size range, this trend reverses. The value of 1/Mwapp decreases with particle mass, while scattering curves increasingly bend. Combination of both tendencies enlarges the fractional uncertainty of 1/Mwapp (and of 1/Lw) drastically. In the size range of our data, ∆a1/a1 increases from 3% to 15% as Lw increases from 300 to 2500 nm. At the same time, the fractional uncertainty ∆a2/a2 of the slope scatters around 5% (Figure 6). Except for a very few data points, the fractional uncertainty of Rg2 does not exceed 20% over a large Lw regime. Fractional uncertainties ∆a1/a1 of 1/Lw, ∆a2/a2 of (〈S2〉z/3!)/Mwapp, and (∆a1/a1 + ∆a2/a2) of Rg2 are summarized in Figure 6 for all three aggregation runs. Obviously, the fractional uncertainties from all three aggregation runs fairly overlap when plotted versus Lw. Only if Lw exceeds 2000 nm, eight data points have a significantly larger error. All of the eight data points belong to the aggregation run induced by the MgSO4 content of 3.3 mM. If we omit those points, a cleared-up set of data remains with an uncertainty of 10% < ∆Rg2/Rg2 < 20%. These refined data, stemming from all three aggregation runs, were used to estimate a reliability range for a. A similar error analysis was performed with the data from the system NaYD/RD, the results of which are shown in Figure 7. For the sake of better clarity, we now confine to a representation of the fractional uncertainty of Rg2. Unlike the system AmYD/RD, fractional uncertainties from different aggregation runs do not overlap when plotted versus Lw. However, all three fractional uncertainties ∆a1/a1, ∆a2/a2, and ∆Rg2/Rg2 for the aggregation run with the highest MgSO4 content by itself do show the same trends as the data in Figure 6. The fractional uncertainties of Rg2 fall between 10% and 20%, gradually increasing from the lower to the higher limit. The uncertainties with the medium content of MgSO4 are persistently larger than 20%, and data from the respective aggregation run will be omitted at this time. Finally, the uncertainties at the lowest MgSO4 content are regarded still to be acceptable with values around 20% for ∆Rg2/ Rg2, although they are somewhat higher than the uncertainties at the MgSO4 content of 3.3 mM.


Figure 7. Fractional uncertainty of ∆Rg2/Rg2 as a function of experimental Lw for the system NaYD/RD. Curves a and b comprise data at [MgSO4] ) 3.3 mM and [MgSO4] ) 1.63 mM, respectively; curve c corresponds to the data at [MgSO4] ) 2.44 mM. Data from the latter aggregation run were not included in Figure 8.

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an uncertainty of 10% with the apparent mass per unit length MLapp. This additional error is not considered by eq 10 but still raises the error in a drastically. Only if Lw exceeds 500 nm, the limiting behavior of qP is clearly established and uncertainty of MLapp can be neglected. Therefore, we do not regard the sharp increase of a at very low Lw as being significant. Finally, we consider the gradual decrease in a with growing wormlike chains. This trend can be denoted as significant within the framework of a 10% uncertainty for Rg2. But if the larger limit of a 20% uncertainty is applied, the trend is not significant and use of a single value for a is justified. However, the reproducibility of the trend with two systems, NaYD/RD and AmYD/RD, justifies a search for potential explanations. As remains to be shown in the following section, a possible explanation may be branching. The Effect of Branching

Figure 8. A plot of persistence length evaluated according to eqs 5 and 6 versus experimental contour length Lw for both dyestuff mixtures AmYD/RD and NaYD/RD at [MgSO4] ) 1.63 mM (O), [MgSO4] ) 2.44 mM (0), and [MgSO4] ) 3.3 mM (4). In both figures, only a data which were evaluated from data with ∆Rg2/Rg2 smaller than 20% are shown. The curves denote theoretical uncertainty regimes based on fixed values of ∆Rg2/ Rg2: (solid line) ∆Rg2/Rg2 ) (20%; (dotted line) ∆Rg2/Rg2 ) (10%.

Having thus established a range of fractional uncertainty between 10 and 20% for the large majority of Rg2 data, an uncertainty of the persistence length a can be evaluated. This was performed according to the following procedure: The theoretical values of Rg2 calculated according to eq 5 with a persistence length of a ) 180 nm and represented as a curve in Figure 5b were enlarged (and reduced) by a fraction of 10 and 20% of their original value. For each of the four resulting sets of Rg2 values, persistence lengths were fitted by means of eqs 5 and 6. The resulting a values are represented as two pairs of curves in Figure 8a,b. The inner pair and outer pair limits the reliability range for a 10% and a 20% error in Rg2 data, respectively. As expected, both reliability bands diverge if 2a approaches Lw because Rg is becoming insensitive to a change in the persistence length. At the same time, bending rod plots of the scattering data of the very first measurements of the aggregation runs with the lowest MgSO4 content may approach but not yet reach the plateau regime, leaving

As was pointed out in the section on the model of broken wormlike stars, introduction of branches into wormlike chains has a similar effect on the scattering curves as an increase in chain flexibility. It is the very change in flexibility which is in fact suggested by the data of Figure 8. Experiments revealed a persistence length of roughly 200 and 220 nm for the systems AmYD/RD and NaYD/ RD, respectively, if the chain length falls in the range of 500 nm < Lw < 1000 nm. As Lw increases from 1000 to 2000 nm, a decreases gradually reaching a value of ca. 150 nm for both systems. Assuming now that this tendency is significant yet the actual persistence length of both systems remains constant at the high level of 200-220 nm, the question arises of how this apparent contradiction can be resolved. Referring to the bending rod plots of broken wormlike chains (Figure 3), the apparent decrease in a may be due to the interpretation of experimental data with the model of linear wormlike chains. In such a case, occurrence of branching points would inevitably be misinterpreted by an increase in chain flexibility. To support this hypothesis, comparison of two theoretical scattering curves with an experimental curve shall be performed as an example. First, an experimental scattering curve is selected with a contour length of Lw ) 2957 nm. This curve falls well within the regime where the persistence length evaluated with eqs 5 and 6 is 150 nm. Two theoretical curves are used to describe the experiment: (i) one curve is calculated with the Koyama approximation for a polydisperse linear wormlike chain with z ) 1 and the chain parameters Lw ) 2957 nm and a ) 150 nm, the latter being extracted by the fit of eqs 4-6 to the experiment; (ii) the second curve is calculated with eq 8 for broken wormlike stars with z ) 1 and 3 arms. Now, a persistence length of a ) 220 nm is used which is assumed to prevail over the entire contour length range of 500 nm < Lw < 2000 nm. All three curves are summarized as bending rod plots in Figure 9. Obviously, both theoretical curves give a satisfactory description of the experimental data. Further refinement of the model of branched aggregates by accounting for a distribution of arm numbers or a variation of arm lengths within an aggregate is not considered to give further insight when applied to the present data. Also, we do not believe that the weakness of Koyama’s approach at qa ∼ 1 interferes with the qualitative conclusions from the above example. Experimental scattering curves with contour lengths significantly smaller than 3000 nm are by far less instructive than the experimental curve chosen for the

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preference to the model of branched worms. We rather put emphasis on the fact that care has to be taken when scattering curves are interpreted in terms of the model of wormlike chains.

Figure 9. Experimental scattering curve from the system AmYD/RD at Lw ) 2957 nm, t ) 926 s, and [MgSO4] ) 3.3 mM in comparison with two theoretical curves: (dotted line) polydisperse linear wormlike chain according to Koyama’s approximation (ref 26) with z ) 1 and a ) 150 nm; (solid line) polydisperse wormlike star according to eq 7 with f ) 3, z ) 1, and a ) 220 nm.

comparison in Figure 9. The reason for this is that it was no longer possible to describe the experimental curve with a pure model star based on the higher persistence length of 220 nm. Use of a theoretical curve from a mixture of 3-arm stars and linear chains became necessary instead. This is most likely caused by the fact that the onset of branching is not abrupt, that is, the fraction of branched chains increases gradually with increasing aggregation time. In conclusion, the model of broken wormlike stars elegantly explains the decrease in a with increasing Lw: As the wormlike aggregates grow in length, branching occurs, which manifests itself in an apparent decrease of a. However, this decrease is at the verge of becoming significant, and to a first approximation, experiments can be described satisfactorily by the model of linear wormlike chains with an averaged value of a. Thus, we do not give

Summary Calculation of scattering curves for broken wormlike stars34 based on Koyama’s approach26 was extended to account for a polydispersity in the overall contour length of the stars. The number of arms, all equal in length, was fixed. Introducing branching centers and/or increasing the number of arms within the stars has a similar effect on the scattering curves as an increase in the flexibility of linear chains. The latter is achieved by simply decreasing the persistence length. Contrary to this ambiguity, evaluation of the mass per unit length from the limiting behavior of bending rod plots remains unaffected by branching a linear chain. By application of the model of combinatorial stars with broken worms to the scattering behavior of dyestuff aggregates, it could be shown that it is difficult to distinguish between a slightly branched worm of higher stiffness and a linear worm of lower stiffness. The simultaneous occurrence of linear aggregates together with aggregates with more than two rays may easily be misinterpreted by a lower persistence length when described by the model of linear wormlike chains. Hence, slightly branched material, if it does occur as intermediates toward network or gel formation, may not be unambiguously identified by scattering methods alone. In such a case, supplementary methods as for example direct imaging by means of electron microscopy become necessary. Acknowledgment. The authors thank Ciba Spezialita¨tenchemie AG for having made available the dyestuffs. LA020143U

Model of Polydisperse Wormlike Stars and Its Application to Dyestuff

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