DISSIPATIVE SELF-ASSEMBLY:
A NOVEL SELF-HEALING MECHANISM FOR FUNCTIONAL
MATERIALS
G. J. M. Koper 1, J. Boekhoven 1,2, W.E. Hendriksen 1, R. Eelkema 1, J.H. van Esch 1
1 Department of Chemical Engineering, TU-Delft, Julianalaan 136, 2628 BL Delft, the
Netherlands – e-mail: g.j.m.koper@tudelft.nl
2 Currently at: Inst. for BioNanotechnology in Medicine, Northwestern University, Chicago, Il,
USA.
Keywords: self-assembly, dissipation, self-healing, functional materials ABSTRACT
Self-assembled systems formed of micelles or vesicles have frequently been discussed as model systems for self-healing materials because their structure is dictated by thermodynamics and hence they quickly restore upon perturbation. In this aspect, they mimic many natural systems such as biological cells. However, in contrast to most synthetic self-assembling systems the natural systems are not equilibrium processes. Attention is therefore now focusing on dissipative self-assembling systems where energy input is required to sustain the self-assembled state. These systems have the potential to adapt themselves and enter into different self-assembled states depending on the rates of environmental reactions whereas their equilibrium counterparts can only assemble or disassemble depending on the environmental equilibrium condition.
Recently, we have constructed some dissipative self-assembling systems using chemical fuels and presently more examples are being worked on. During this presentation some important aspects of these systems will be discussed in relation to their capabilities of being self-healing.
1. INTRODUCTION
Many functional self-healing materials are synthesized by means of a self-assembly process. Examples are the self-healing rubber of Ludwik Leibler [1], the supramolecular polymers of Bert Meijer [2], the nanofiber forming peptide-amphiphiles of Sam Stupp [3], and the self-healing hydrogels of Takuzo Aida [4]. From a physical point of view, the linear self-assembly process of Stupp’s peptide-amphiphiles is the simplest and we shall use it to illustrate some of the important aspects; a cartoon is given in Figure 1a. In addition, it also represents the simplest class of supramolecular polymerizations as reviewed by Alberto Ciferri [5].
The rate of formation of an aggregate of N monomers is given by
1 1
N f
r k x (1)
where x1is the mole fraction of monomers and k1 the forward rate constant. The
Figure 1: Linear self-assembly of units with binding sites on either side (a), size distribution for total monomer mole fractions 1, 10 and 100 times the cgc (b), and
N b N x r k N (2)
where xN is the mole fraction of monomers aggregated into aggregates consisting of Nmonomers, so called N -aggregates. When we assume equilibrium, the associated
equilibrium constant K – per monomer – is defined as
1
N kN
K k
(3)
With a total monomer mole fraction xt one finds the characteristic,
exponentially-tailed size distribution, see Figure 1b, as predicted using the simple mean field model by Cates and Candau [6] with an average degree of polymerization, see Figure 1c, scaling as [7]
1/2
t
N x (4)
The network structures of Leibler [1] are from a physical point of view much more complex, see Figure 2, as they do not only consist of di-functional units, that are responsible for linear aggregation, but also of tri-functional units with which branching junctions are formed. With these building blocks the resulting structure becomes cross-linked as sketched in Figure 2. A geometrical analysis, not surprisingly, yields a length distribution of the mesh size L that is again exponentially tailed and an average mesh size that scales also algebraically with total mole fraction albeit with an exponent of 0.56 which is slightly larger than the 0.5 for the above described linear case [8]. The case of threefold junctions – as present in Leibler’s system – is interesting because the predictions both by Drye and Cates [8] and later by Zilman and Safran [9] indicate the possibility of a phase transition. Experimentally it is indeed found, that crystallization is hampering the synthesis of such systems [1]. Aida’s system of clay particles that are interconnected by dendritic binders are – from a physical point of view – a variation of Leibler’s system where the average functionality of the branch points is typically larger than 3 and hence the behaviour is less interesting as no phase transitions are to be expected.
A crucial aspect of the above described processes is that it is in equilibrium which means that the forward rate rf (eq.(1)) and the backward rate rb(eq.(2)) are balanced
and this indeed is the condition from which the relation for the equilibrium constant emerges, see eq.(3). It is important to realize that this condition is to be taken literally, meaning that indeed there is a constant exchange of monomers between the aggregates themselves and with their environment where there are only isolated monomers. It is generally accepted, that there are two time scales associated with these aggregates. One is associated with the exchange of single monomers with the aggregates. It is relatively short and fully determined by the diffusion coefficient of the monomer. The slow time scale is associated with the formation process of one complete aggregate and roughly equal to the fast time scale times the average aggregation number. Hence, self-assembly processes are dynamic by nature and not static as sometimes suggested [10]. This dynamic aspect is what makes self-assembling systems an interesting option for self-healing materials. The only disadvantage is that when the system consists of more bulky monomers, the time scales become inherently longer. Therefore, even though self-assembling systems can be tuned by environmental conditions, the response times are relatively long to the extent that some systems do not reach equilibrium at all.
2. A SYNTHETIC DISSIPATIVE SELF-ASSEMBLING SYSTEM
The equilibrium self-assembling systems described above are much less dynamic than their natural counterparts and it is interesting to see why this actually is the case. It is clear that the natural systems are not in equilibrium and require transfer of energy to operate: in biological systems it is often the ATP hydrolysis that conveys the necessary energy [11]. Examples from Nature are networks built from for instance microtubules or compartments such as mitochondria.
To investigate this idea, we chose a synthetic system that has recently been put forward by our group [12,13], see Figure 3. The self-assembly utilizes a low molecular mass hydrogelator, dibenzoyl-(L)-cystine (DBC) in aqueous solution at a pH above its pKa value of 4.5. Under these conditions DBC remains isotropically in solution. Methylation of one of the carboxylic groups results in the formation of fibres and this process can be detected mechanically and optically, using for instance rheometry and turbidity measurements respectively. Crucially, the methylated DBC molecules are chemically unstable and hydrolyse at a pH-dependent rate. Hence, by tuning the pH and the methylation reaction rate the self-assembly dynamics of the gel can be controlled. The methylation reaction is not spontaneous and therefore it is coupled to a reaction that is spontaneous which in the present case is the conversion of dimethyl sulphate (DMS) into monomethyl sulphate (MMS-). In actual fact, also the hydrolysis reaction is a coupled reaction that turns the methylated DBC back into DBC itself, see Figure 3a.
Let us now describe the phenomenology of dissipative self-assembly for this system. As soon as fuel is added to the system and the methylation reaction starts, the concentration of self-assembling monomers, the methylated DBC molecules, increases, see Figure 3b. When the critical gelation concentration (cgc) is exceeded the storage modulus starts to increase, see Figure 3c, which indicates that gel formation has started. As soon as the total concentration of monomers – free as well as aggregated – increases, their destruction also sets in by means of the hydrolysis
Figure 3: Dissipative self-assembly, schematic of the process (a), total concentration of monomer, the methylated DBC (b), storage modulus of the gelled system (c), and
relative recovery after perturbation (d). a)
b)
c)
reaction. Initially, the formation rate exceeds the destruction rate and the monomer concentration increases resulting in an increasing storage modulus. At the end of the experiment, the destruction rate dominates and the gel breaks down leading to a decreasing storage modulus. In the intermediate regime, where formation and destruction roughly balance, the storage modulus remains reasonably constant. A closer look at the dynamical behaviour of this system reveals, that the maximum in the total monomer concentration occurs before the maximum storage modulus is reached. This most likely is due to the additional time scale associated with fibre formation and break-down and of course the strongly non-linear dependence of gel strength on fibre concentration.
The most interesting question is now, whether this system possesses self-healing properties, that is how does the gel behave after mechanically induced perturbations? Typically, gels formed by self-assembly of low-molecular-mass gelling agents are not capable of self-healing to their original strength, mainly because of the slow assembly and disassembly dynamics of the system at ambient conditions, although exceptions are described [4]. When a gel is subjected to mechanical perturbations the long fibres break into smaller fragments and because of the slow assembly dynamics the fibres are not able to re-assemble and regenerate the former gel properties. The continuous formation and destruction of molecular building blocks in the present out-of-equilibrium system is expected to alleviate this.
In Figure 3d the relative recovery, the ratio of the maximum storage modulus after and before perturbation, of the gels is plotted versus time. As expected, the recovery is very weak in the final regime where break-down of the monomers dominates. In the intermediate regime, where the formation and destruction more or less balance, the recovery is reasonable. The recovery is very good when the formation rate of monomers is dominating.
3. DISCUSSION
The process described here is a non-equilibrium processes in the thermodynamic sense that the associated Gibbs energy change is negative, it runs spontaneously, and that if this Gibbs energy change is not converted to work it is dissipated by the environment in the form of heat [14]. It is therefore that the non-equilibrium form of self-assembly as discussed above is called dissipative self-assembly even though it would in principle still be possible to convert the Gibbs energy change into work which is not immediately lost. In a recent study we evaluated the amount of work that is lost in a dissipative self-assembling system. For the system presented above, the lost work is largely due to the driving reactions, the methylation and the hydrolysis, and amounts to 5.5 W/L solution [15].
Despite the promising result, there are still quite some questions one may ask with regards to this system. In particular, the formation and break-down of the aggregates in relation to the formation and destruction of the monomers is an interesting issue that warrants further investigation. The relation between the structure of the system and its function, its mechanical response, is a classical one. The fact that the structure of the system is dynamical might actually be beneficial in the sense that it allows for a much more detailed study of the behaviour as one may change the structure at experimental time scales.
Dissipative self-assembling systems may find their own fields of application. One of them is surely where long-lived stable moulds are needed. During their life time, the moulds need to be self-healing, but after use they should be easy to discard. In fact, all applications where the constituting materials need to be removed easily after use come within reach which provides a great potential for all applications that need to be running in an environmentally friendly manner.
Other potential applications – not necessarily self-healing – become clear when it is realized that the monomer destruction reaction need not run simultaneously with the monomer destruction reaction. In other words, with a running formation reaction one creates a self-assembling system that remains in place until the destruction reaction is run. Apart from the fact, that the system then acts as an energy store such as a battery where this energy can be reclaimed when the destruction reaction for instance is done electrochemically.
ACKNOWLEDGEMENTS
The authors acknowledge the Netherlands Organisation for Scientific Research (NWO) for financial support.
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