Introduction

Block copolymers are macromolecular systems where (linear) polymer P i (for short, “block i”), differing from each other by their chemical architecture, are linked together by their ends thanks to the ingeniousness of polymer chemists, which results in linear architectures P1-P2-…-P n in the general case of a n-block copolymer. Because each P i is a macromolecule, the general argument stemming from Flory’s estimate of the mixing entropy [1] suggests that blocks are mutually incompatible. Phase separations are indeed commonly observed in P1 / P2 / … / P n mixtures at a macroscopic scale but, owing to the chemical links between adjacent blocks, it is no longer possible with block copolymers. Still, a local segregation remains possible and, as is well known from long ago, domains enriched in component P1, P2, etc. may appear under favourable circumstances [2, 3]. The interesting point is that one characteristic size of such domains is, roughly speaking, in the order of the radius of gyration of polymer P i for domains enriched in block P i , that is to say a supramolecular scale. And domains may be stacked in a periodic fashion in space, leading—depending on their shapes—to 1D, 2D or 3D large-scale structures. Lamellar, tubular or globular morphologies are typically observed, in particular for the simplest case of diblock copolymers, as early revealed by electron microscopy for instance [4].

Large-scale structures may also be present when block copolymers are mixed with solvents, in particular if a solvent is “good” for some blocks, but “bad” for the others, as the polymer plays then the role of an amphiphilic (macro) molecule. Self-organised micellar liquid or crystalline phases, as well as lamellar or columnar liquid-crystalline phases, are found, and their characteristic sizes are indeed much larger than for most of their surfactant-based counterparts [5, 6]. Quite a common problem, however, especially in the bulk, is—quoting from a 40-year-old review [3]—that:

[ …] the structure may very well remain ill-defined over a long period of time after any treatment of the sample: heating, cooling, evaporation of the solvent, dilution, etc.

Indeed, because polymers are easily entangled, viscosity is usually high in the bulk which, combined with the large scales exhibited in block copolymer systems, leads to a generically slow dynamics. In the presence of long-range order, dynamic properties are in addition tagged by a characteristic anisotropy [7], but such a remarkable feature apparently remained ignored for a while by some experimentalists. As a matter of fact, for experimentally studying dynamic properties, dynamic light scattering (DLS) emerged long ago as a powerful technique in disordered and dilute block copolymer solutions [6], but led to somehow ambiguous results in more concentrated or ordered systems [8–13]. This may have motivated a remark written 20 years ago in another famous review [14]:

[…] interpretable results have only been obtained in the disordered state, up to the ODT […]

As shown below, in “Dynamic light scattering” section, some progress has been made in the meantime, regarding dynamics in the lamellar phase mostly. And “Small-angle neutron and x-ray scattering” section is devoted to recent illustrations of structural studies in long-range-ordered phases with not so “ill-defined” structures, after all.

Dynamic light scattering

Dynamic light scattering has gradually become a technique of choice for studying polymer solutions, following the pioneering contributions in the field by Pecora [15, 16] from the theoretical side and by Cummins et al. [17] as experimentalists. An early contribution by P. Štěpánek has been the study of the homopolymer coil–globule transition with this technique [18] and, soon afterwards, the dedicatee was expert enough in measuring Doppler shifts for co-authoring a comprehensive review “in the field of synthetic polymers and colloidal particles” [19], where however block copolymers were not central—yet. Block copolymer micelles were specifically studied not much later, in Ref. [20, 21] for instance. At that time, the systems were still kept quite far from their order-disorder transition, in their disordered phase. Refinements in the analysis of dynamic light scattering data were also extensively developed in apparently simpler polymer systems, viz. Θ solutions of polystyrene [22], and it was then firmly established that “the relaxation time distributions increase in complexity with increasing polymer concentration”.Footnote 1

The latter comment was found for a while to apply to the case of diblock copolymer in concentrated solutions as well: “Slow” and “fast” modes were found in Ref. [8], “with one of the frequencies characterising the peculiar dynamics of internal motion of diblock copolymer chains”, and—almost simultaneously—in Ref. [9] also. As this latter study was then more extensive—as far as copolymer concentrations in the neutral solvent were concerned—and complemented by small-angle x-ray scattering, the role of order (specifically, the self-assembly into lamellar micro-domains) could be suggested, at least in qualitative terms, even if the insightful analogy with “lamellar bilayer membranes” was unfortunately not investigated deeply enough [9]. On the other hand, depolarised dynamic light scattering was explicitly associated with the presence of hexagonal or lamellar long-range orders in Ref. [10] and [11], respectively, but time has not yet come to the study of oriented long-range-ordered block copolymer systems, even though such sample preparation techniques allowing to distinguish between appropriate projections, say q z and q ⊥, of the scattering wave vector q have already been successfully applied to thermotropic [23] and lyotropic liquid crystals [24–26].

As far as bulk diblock copolymer systems were concerned, a somehow confuse situation was also prevailing, either when the focus of DLS experiments was disordered [27–29], or ordered hexagonal [30] or lamellar [31] systems, or even systems with other symmetries, including gyroid and body-centred morphologies [32, 33]. Still, depolarised DLS was again rather clearly associated with the occurrence of long-range order, as in Ref. [34]. Besides, in this latter contribution, the undulation mode of smectic A liquid crystals was casually mentioned, but the lack of oriented samples was definitely impeding further progress.

It should be noted that (mechanically) oriented samples of diblock copolymer were already being produced at the time and studied, by means of forced Rayleigh scattering as in Ref. [35, 36], for instance. Analyses in terms of anisotropic properties in lamellar or hexagonal systems were also proposed in pulsed field gradient NMR where the need for oriented samples was circumvented through the clever trick of an isotropic averaging of the expected dynamics, as in Ref. [37–39]. However, compared to DLS, these latter techniques gave no direct insight into collective dynamic properties.

The synthesis of a nearly symmetric poly(dimethylsiloxane)-b-poly(ethylene-co-propylene) (for short, PDMS-PEP) with molar mass M n = 6.3 kg/mol and order-disorder transition temperature T ODT = 64 ∘C [40], both conveniently low for sample handling, was a significant step for obtaining an oriented (lamellar) sample suitable for dynamic light scattering experiments. With the proper control of both the modulus q and projection q z of the scattering wave vector q, it became possible to evidence the so-called undulation mode, an elastic distortion characteristic of smectic A liquid crystals when q z = 0, i.e. q parallel to the smectic layers. Moreover, from the expected dispersion relation anisotropy—the undulation mode merging into “second sound” as q becomes oblique with respect to the smectic layers [41]—the mode relaxation frequency in a disoriented sample of the same material, measured previously [34], could be predicted from the relaxation frequency of the undulation mode measured in the oriented sample, namely (for q z ≪ q ⊥):

$$ {\Gamma}=\frac{K}{\eta}q_{\perp}^{2}\left[\left( \frac{q_{\perp}}{q}\right)^{4}+\frac{{q_{z}^{2}}}{\lambda^{2}q^{4}}\right] $$
(1)

with K the splay modulus, λ the so-called smectic penetration length and η a shear viscosity (\(q_{\perp }\equiv \sqrt {q^{2}-{q_{z}^{2}}}\), of course). A quite convincing agreement was then obtained [42], perhaps one of the great achievements by P. Štěpánek in his career-long quest of modes in polymer systems. “Interpretable results” were at last obtained by DLS in the (lamellar) ordered state, but only for one, exquisitely tailored, diblock copolymer system. Fifteen years later, it does not seem that similar achievements have been reached with 2D hexagonal or 3D cubic phases of block copolymers. Admittedly, orienting the samples—in the cell, but also with respect to the scattering coordinates—is not an easy task, and the theoretical framework for interpreting the results may be found a bit frightening, as shuffling through Ref. [43] (devoted to 2D hexagonal phases in surfactant-based systems) could convince undaunted readers.

Applying DLS to block copolymer micelles, i.e. in the presence of a low molecular weight solvent, is presumably easier, generally speaking, but also rewarding in association with fluorescence correlation spectroscopy, in particular for accurately determining the (usually very low) critical micellar concentration (CMC) of such systems, as well as the hydrodynamic size of the emerging micelles close to the CMC, and P. Štěpánek contributed to such studies [44]. Besides, even if block copolymer micelles may appear as very “simple” objects, from a geometrical point of view, being therefore quite routinely characterised using DLS, their detailed structures may occasionally be difficult to describe in intuitive terms, as in the conspicuous case of a diblock copolymer of styrene and ethylene oxide forming micelles in mixtures of 2,6-dimethylpyridine and water over a broad solvent composition range, although neither polystyrene nor polyethylene oxide homopolymers are soluble in such mixtures [45]. Resorting to small-angle neutron or x-ray scattering techniques (“Small-angle neutron and x-ray scattering” section) as tools complementary to dynamic light scattering then cannot be avoided, as already illustrated in Ref. [45] or other relevant contributions by P. Štěpánek such as, for instance, Ref. [46–51]. When the polymer material is no longer a block copolymer, see Ref. [52], precisely describing the nanoparticle structure is even a tough task, and work is still in progress [53].

Small-angle neutron and x-ray scattering

As is well known, “static” (or elastic) scattering techniques, including light of course, have been used for a long time for obtaining valuable structural information at various levels of detail, in particular

in order to examine […] the attenuation undergone by the primary light on its passage through a medium containing small particles, as dependent upon the number and size of the particles

quoting Lord Rayleigh himself [54].

The question of particle sizes in the context of dilute polymer solutions was addressed by P. Štěpánek with DLS, as already mentioned in “Dynamic light scattering” section, but also with conventional elastic light scattering in the same contribution [18]. The focus was “particle” size, and the technique static light scattering alike in Ref. [55]—admittedly extending a bit the concept of particle to the polymer “blob” of semi-dilute polymer solutions [56, 57]—or in Ref. [20, 21], with triblock copolymer micelles (and a more conventional meaning as regards the “particles”).

Small-angle scattering techniques using neutron, Ref. [58], and x-ray, Ref. [9], appear not much later, even though in both cases no explicit reference to Lord Rayleigh’s particles were made, as the focus was then, respectively, critical fluctuations in polymer mixtures and long-range order in a concentrated diblock copolymer solution, and not the polymer micelles themselves. Efforts were still about determining the long-range order of block copolymer domains in Ref. [59], with, however in this contribution, a brief discussion about “suppression of higher-order [Bragg] reflections” because of “micellar form factor [having] minima”.

Assessing the structure simultaneously in terms of particle (micro segregated domain) shapes and long-range order was nevertheless quite a standard objective in the 1970s—see, for instance, Ref. [3]—and early 1980s, as exemplified for instance, as far as block copolymers are concerned, in Ref. [60] or at about the same time in a noticeable contribution, Ref. [61], by the famous Kyoto team. Not only Bragg peak positions were recorded and elementary crystallographic arguments used for differentiating, e.g. various flavours of 3D cubic order, but simple geometric models in association with dilution laws and a rudimentary analysis of “intraparticle interference” (or form factor) maxima were also introduced to very efficiently disclose the actual structures of block copolymers (in solution or in the bulk) in cases where, owing to the intrinsic features of the SAXS data, a partial analysis (restricted to Bragg peak positions as in Ref. [59], for instance) would have been less conclusive. Interestingly, the authors of Ref. [61] make use in their analyses of the few first values of the secondary maxima of \(\left (\sin x-x\cos x\right )^{2}/x^{6}\equiv {j_{1}^{2}}(x)/x^{2}\) and \({J_{1}^{2}}(x)/x^{2}\) without even explicitly mentioning that these Bessel-related functions correspond to, respectively, the homogeneous sphere or cylinder form factors in reduced units—and without specific worries about the adverse impact on their reasoning of form factor minima being actually equal to 0 (a mathematical property already taken into account, albeit qualitatively, in Ref. [62]).

The underlying idea is that, similarly to the case of a liquid (i.e. disordered) phase of identical spheres where the dependence in wave vector modulus q of the scattered intensity I is expressed as the product of a form (“intraparticle interferences”) and a structure (“interparticle interferences”) factor, I = P(q) × S(q), see Ref. [49, 50, 63] for older and more recent block copolymer examples; the same factorisation remains essentially valid for the randomly oriented (or “powder”) sample of a long-range-ordered phase of well-defined particles. In mathematical terms:

$$ \frac{1}{4\pi}\int d^{2}\!\mathbf{n}\:P(q\mathbf{n})\times S(q\mathbf{n})=P_{\text{iso}}(q)\times S_{\text{iso}}(q) $$
(2)

with n a randomly oriented unit vector, P(q) the form factor of a (possibly anisotropic) particle with a fixed orientation with respect to the single crystal built by stacking those particles in a 1D, 2D or 3D fashion in space, S(q) being the structure factor associated with the corresponding long-range order.

Whereas Eq. 2 is rather obvious for isotropic particles, that is to say spherical domains, it requires some attention with 1D ordered lamellae [64–67] or 2D ordered cylinders. When the simplifying assumptions allowing its derivation are valid, Eq. 2 is at the basis of the quantitative analysis of the whole scattering curve, with its Bragg and diffuse components taken into account on the same footing. Following Ref. [66], early examples of the method with a diblock copolymer sample may be found in Ref. [68], with however an ad hoc structure factor, or in Ref. [69] where, even if no attempts were then made to model the (complex) face-centred or body-centred cubic structure factors, the interesting idea of describing the form factor in the concentrated solution in terms of the one measured in a dilute solution was clearly present. Among the merits of such an approach, the suggestion that the dispersity Ð in particle sizes is a very relevant cause for not observing the form factor minima equal to 0 in the scattering data. But the mechanisms at play in the self-association process may, or may not, lead to significant changes in particle shape or size as polymer concentration increases and, therefore, to a potential weakness in an analysis relying too much on this idea.

A direct extraction of the form factor from the small-angle scattering data in concentrated block copolymer systems, with the help of contrast variation— S iso(q) assumed unaffected—or assuming, as a characteristic property of soft matter, that S iso(q) quickly reaches its asymptotic limit, namely 1, may be found in Ref. [70] and [71–73], respectively. As a matter of fact, the apparently paradoxical idea that S iso(q) becomes constant and ≈ 1 for not too high q values, implying that small-angle scattering curves contain in practice important information about the particle form factor even though those particles are strongly correlated, had already been experimentally demonstrated a few years ago by small-angle neutron scattering in the so-called sponge (or L3) phase found in some dilute surfactant–solvent systems [74, 75]. The particles, somehow isotropically stacked in the sponge phase (as theoretically described in Ref. [76]), were indeed found identical to the familiar surfactant bilayer, i.e. the building unit in the 1D ordered lamellar phase of lyotropic smectics. In this latter case, an explicit fit of most of the SANS or SAXS scattering curves to a simplified model of stacked bilayers could be proposed [67].

Particles were not yet fully described as such in Ref. [77], but taking into account the property S iso(q) ≈ 1 allowed to extract information about chain statistics in the “corona” of the solvent-swollen, self-assembled block copolymer building blocks of the 2D or 3D long-range-ordered structures investigated and to show later that chain statistics is not strongly affected by applying an electric field to the system [73]. It had thus taken half a decade to reach a nearly complete description of the whole scattering curve, with both intraparticle and interparticle structural contributions fully and quantitatively accounted for in terms of a body-centred structure of spheres (with some dispersity Ð) [78].

A still pending question for interested readers would be to evaluate the respective merits of the methods proposed in Ref. [71] and [78] for computing the appropriate model structure factors and fitting to real data.

Conclusion

Tremendous progress has been made in about six decades in producing and studying block copolymers, but a remarkable stability still prevails since the (first, perhaps) report concerning such systems by A. Skoulios and others [5] in August 1960: Scattering techniques have been central in establishing and elucidating the conspicuous properties of these fascinating representatives of soft matter. The road map for their structural study, as expressed by A. Skoulios again in Ref. [3], has been quite successfully followed along the years, ease of access to small-angle neutron and x-ray scattering techniques (including at synchrotron facilities in the latter case) vastly contributing to such successes. The above-mentioned road map did not include dynamic light scattering techniques (and its more recent extension to x-ray photon correlation spectroscopy on synchrotron sources), nor the neutron spin-echo technique, but they have been widely used and also led to a significant progress in the field. Of course, reciprocal space and Fourier transforms are not the ultimate tools: Other techniques proved useful in studying block copolymers, rheology among many others, with a farseeing comment in this respect formulated quite early [62]. Though being, as it may now seem, essentially a Fourier disciple, P. Štěpánek also contributed to advances in block copolymer knowledge outside (q,ω) space—but this is another story.