Amphiphilic molecules contain at least two structural units that thermodynamically repel each other. Since the two incompatible blocks are covalently bonded into a single molecule, they cannot macroscopically phase separated but, instead, self-assemble into spatially modulated structures whose characteristic length scale is dictated by the molecular extension. Typical examples include the self-assembly of lipid molecules, which are comprised of a hydrophilic, polar head and a hydrophobic tail, into bilayer membranes or synthetic block copolymers, which are comprised of two incompatible flexible chain molecules that are joined at their ends and self-assemble into periodic microphases. Despite the differences in the chemical nature of the constituents and the type of interactions, both – biologically relevant lipids as well as synthetic block copolymers – spontaneously form similar structures (e.g., lamellar sheets or wormlike micelles). The structure formation is dictated by the universal competition between the free-energy cost of the interface between the incompatible components and the entropy loss of arranging these molecules uniformly in space.
This delicate balance gives rise to minuscule free-energy differences between different morphologies (on the order of a fraction of the thermal energy scale kT per molecule), and there exist many competing metastable structures (alternate periodic arrangements, defect structures like dislocations, or localized structures like hydrophobic bridges between lipid membranes). This feature is corroborated by the protracted annealing times required to observe well-ordered morphologies in block copolymers or the requirement of specialized proteins that provide the free energy required to overcome the barrier in e.g., pore formation, fusion and fission of membranes. In fact, the complex, rugged free-energy landscape of self-assembling amphiphiles has been likened to that of glass-forming materials. The morphology often does not reach the thermodynamically stable state of lowest free energy but, instead, becomes trapped in a metastable state. By exploring these metastable states and the free-energy barriers that separate them, one can either reproducibly trap the system in desired non-equilibrium morphologies  or accelerate equilibration of block copolymer structures  or control collective changes of membrane topology involved in cellular transport processes [3,4].
Morphological transformations of amphiphilic, soft-matter systems involve the cooperative rearrangement of many molecules on time and length scales ranging from milliseconds to minutes and nanometers to micrometers for lipids and polymers, respectively. These scales are challengingly small for experimental imaging techniques yet too large for atomistic modeling. Since the transformations often involve highly bent interfaces or strongly stretched molecular conformations, also phenomenological continuum models cannot accurately capture them. In turn, coarse-grained models that only incorporate the relevant degrees of freedom are well suited to explore the universal behavior of amphiphilic structure formation and provide direct insights into the kinetics as well as the free-energy landscape. The following two examples illustrate recent progress.
Process-directed Self-assembly of Block Copolymers on chemically guiding Patterns
Block copolymer lithography directs the self-assembly of block copolymers in thin films by sparse, lithographically fabricated, chemical or topographical substrate patterns into dense nanostructures with a critical dimension of a few nanometers. Applications in microelectronic industry require an extraordinarily low defect density of less than 1 defect per 100cm2. Computer simulation and self-consistent field theory demonstrated that the excess free energy of a defect is several 100 kT, making the probability that thermal fluctuations generate defects in an initially ordered structure vanishingly small. However, since defects are observed in experiments, they must arise during the self-assembly process and, thus, it is important to understand the kinetics of self-assembly and defect annihilation [5,6].
Typically the kinetics of self-assembly can be divided in two stages (see Fig. 1 bottom row) : In the initial stage, the homogeneous structure after solvent evaporation is instable , and local domains of incompatible blocks form. This spinodal microphase separation is directed by the lithographic substrate pattern, which imparts overall long-range orientation and registration onto the copolymer morphology, and it lasts the time, τ, a copolymer needs to diffuse its molecular size. The bottom left snapshots of Figure 1 reveal that the morphologies in this early stage are riddled with defects. In the second stage [2,5,7] – defect annihilation and grain formation – defects move in response to long-range strain fields, they collide and may annihilate.
Computer simulations provide insights into defect motion and annihilation mechanisms. Using computer simulations of a highly coarse-grained soft polymer model and self-consistent field theory, we have studied dislocation defects in lamella-forming diblock copolymers [5,7]. Dislocations can easily move along the stripe directions (climb motion) and dislocations with opposite Burgers vectors attract each other. This attraction – Peach-Koehler force – is dominated by near-field and boundary effects, and we find that defects attract each other with a distance-independent force. The dislocation motion critically depends on the distance between the cores perpendicular to the stripes. Upon collision they may spontaneously annihilate (Fig. 1 bottom, lower row χN=20) or form long-lived tight dislocation pairs or disclinations (Fig. 1 bottom, upper row χN=30).
The removal of a metastable tight dislocation pair is a thermally activated process, and we have studied the detailed mechanism by the string method . This computational technique allows us to determine the pathway of defect annihilation and the concomitant free-energy barriers without prior assumption of a reaction coordinate. The top left graph of Figure 1 depicts the free-energy change along the path, α, for different strengths, ΛN, of the chemical guiding pattern. It consists of sequential breaking and re-joining of connections and, although the starting and ending morphologies are quasi-two-dimensional, a proper description of the transition state requires three-dimensional calculations. The top right graph presents the excess free energy of defects and the largest barrier along the defect annihilation path as a function of incompatibility, χN. The defect free energy increases approximately linearly with χN and extrapolates to zero around the order-disorder transition, χNODT=10.5. The barrier of defect annihilation also linearly depends on χN but it vanishes around χN*=18 on an unpatterned surface, ΛN=0, and we can shift this boundary to larger incompatibilities, χN*=24, by increasing the selectivity of the chemical guiding pattern. Thus there is an optimal window of incompatibility, 10.5 < χN < χN*, where defects spontaneously annihilate yet the probability to create defects by thermal fluctuations is vanishingly small. Indeed, the bottom of Figure 1 demonstrates that reducing the incompatibility below χN* for a short initial period can dramatically improve the directed self-assembly .
Two Stages of Dynamin-mediated Fission of Membrane Tubes
The structuring of living organisms into cells and sub-cellular compartments is maintained by lipid bilayers undergoing frequent but carefully regulated topological changes [8,9]. During membrane fission a continuous hourglass-shaped membrane tube is divided into two separate bilayers facing each other. Like its reverse, membrane fusion, the intermediate stages of this process involve energetically unfavorable, highly bent morphologies and dynamin proteins provide free energy to substantially deform the lipid bilayer. In the course of fission, dynamin proteins form helical assemblies around the membrane tube and undergo a conformational change upon addition of GTP (Guanosine triphosphate) causing simultaneous constriction, elongation, and twisting [8,9].
Computer simulations of coarse-grained models can explore the role of the conformation changes of the fission protein and yield insights about the sequence of morphological changes. As a first step we represented the PH (Pleckstrin Homology) domain of the large dynamin protein as amphiphilic disk that shallowly inserts into the membrane con- stricting the membrane tube and locally inducing positive curvature (see Fig. 2) [8,9].
The fission process is divided into two distinct stages: First constriction and curvature gives rise to flickering states (Fig. 2 left), where the lumen of the membrane tube reversibly opens and closes, and eventually induces the transition from a membrane tube to a metastable hemi-fission intermediate – a worm-like micelle that is the analog of the stalk in membrane fusion. Tailoring the position of the PH domains we can induce tilt and curvature, thereby reducing the required constriction and facilitating the formation of the hemifission intermediate .
The hemifission intermediate (Fig. 2 right) is surprisingly (meta)stable . Radial constriction of the dynamin scaffold alone is not sufficient for the hemifission intermediate to spontaneously rupture and complete the fission process. Our simulations indicate that, instead, disassembly of the dynamin scaffold and axial tension may facilitate the final severance of the hourglass-shaped bridge. The detailed pathway from hemifission to fission and the role of dynamin, however, remains yet to be explored.
These examples illustrate that the combination of meaningful coarse-grained models for soft matter, efficient simulation techniques, and computational resources provided by supercomputing centers now enable us to explore the complex free-energy landscape of collective processes in self-assembling materials. Ideally these insights will allows us to design processes  – i.e., time protocols of thermodynamic control parameters (pressure or solvent properties) or localized stimuli imparted by functional molecules – that reproducibly direct the collective kinetics into desired morphologies. Such a process-directed self-assembly will allow access to a plethora of non- equilibrium, metastable structures. This investigation clearly is in its infancy. Whereas the examples indicate its usefulness in two different contexts, there remain many open questions related inter alia to a proper choice of the relevant collective order parameter(s), the relation between the molecular dynamics of individual molecules and the collective kinetics of the morphology, as well as the type of the time-dependent external control or internal stimuli-response.
I have benefitted from stimulated and enjoyable collaborations with Israel Barragan Vidal, Juan de Pablo, Fabien Léonforte, Vadim Frolov, Marc Fuhrmans, Su-Mi Hur, Weihua Li, Paul Nealey, Juan Carlos Orozco Rey, Sandra Schmid, Yuliya Smirnova, Dewen Sun, Ulrich Welling, and Guojie Zhang. Financial support has been provided by the DFG under grants Mu1674/12, Mu1674/14 and SFBs 803, 937, as well as the FP7 project CoLiSA.MMP and the Volkswagen foundation. Computing time at the Neumann Institute for Computing, Jülich, as well as the HLRN Hannover/Berlin and the GWDG Göttingen is gratefully acknowledged.
 Müller, M., Sun, D.W.
Directing the self-assembly of block copolymers into a metastable complex network phase via a deep and rapid quench, Phys. Rev. Lett. 111, 267801, 2013
 Li, W.H., Müller, M.
Defects in the self-assembly of block copolymers and their relevance for directed self-assembly, Annu. Rev. Chem. Biomol. Eng. 6, 187, 2015
 Smirnova, Y.G., Fuhrmans, M., Barragan Vidal, I.A., Müller, M.
Free-energy calculation methods for collective phenomena in membranes, J. Phys. D: Appl. Phys. 48, 343001, 2015
 Fuhrmans, M., Marelli, G., Smirnova, Y.G., Müller, M.
Mechanics of Membrane Fusion / Pore Formation, Chem. Phys. Lipids 185, 109, 2015
 Li, W.H., Nealey, P.F., de Pablo, J.J., Müller, M.
Defect removal in the course of directed self-assembly is facilitated in the vicinity of the order-disorder transition, Phys. Rev. Lett. 113, 168301, 2014
 Hur, S.M., Khaira, G., Ramirez-Hernandez, A., Müller, M., Nealey, P.F., de Pablo, J.J.
Simulation of defect reduction in block copolymer thin films by solvent annealing, ACS Macro Letters 4, 11, 2015
 Müller, M., Li, W.H., Orozco Rey, J.C., Welling, U.
Defect annihilation in chemo-epitaxial directed assembly: Computer simulation and self-consistent field theory, MRS Proceedings 175, mrsf14-1750-kk03-05 (2015)
 Fuhrmans, M., Müller, M.
Coarse-grained simulation of dynamin-mediated fission, Soft Matter 11, 1464, 2015
 Mattila, J.-P., Shnyrova, A.V., Sundborger, A.C., Rodriguez Hortelano, E., Fuhrmans, M., Neumann, S., Müller, M., Hinshaw, J.E., Schmid, S.L., Frolov, V.A.
A hemi-fission intermediate links two mechanistically distinct stages of membrane fission, Nature 524, 109, 2015
contact: Marcus Müller, mmueller[at]theorie.physik.uni-goettingen.de