## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Binary nanoparticle superlattices of soft-particle systems

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved June 26, 2015 (received for review March 9, 2015)

## Significance

The phase diagram of a system of two species of particles with different diameters interacting with a soft (

## Abstract

The solid-phase diagram of binary systems consisting of particles of diameter *p* = 12 power law is investigated as a paradigm of a soft potential. In addition to the diameter ratio *γ* that characterizes hard-sphere models, the phase diagram is a function of an additional parameter that controls the relative interaction strength between the different particle types. Phase diagrams are determined from extremes of thermodynamic functions by considering 15 candidate lattices. In general, it is shown that the phase diagram of a soft repulsive potential leads to the morphological diversity observed in experiments with binary nanoparticles, thus providing a general framework to understand their phase diagrams. Particular emphasis is given to the two most successful crystallization strategies so far: evaporation of solvent from nanoparticles with grafted hydrocarbon ligands and DNA programmable self-assembly.

Arrangement of nanoparticles (NPs) into structures with long-range order encompasses a fundamental new type of materials with potential revolutionary applications in optics, photonics, catalysis, or novel fuel energy sources, just to name a few. Over the recent years there has been a spectacular success in the assembly of nanoparticle superlattice (NPS), with the two most successful strategies consisting of evaporation of a solvent from NPs with grafted hydrocarbon chains (1⇓⇓–4) [solvent evaporation (SE) systems] or the programmable self-assembly of DNA grafted NPs (5⇓–7) in water (DNA systems).

Although there are different models available to investigate DNA systems (8⇓⇓⇓⇓–13), studies of SE systems have been almost (except, for example, in ref. 14) exclusively based on the hard-sphere (HS) model, following the pioneering work of Murray and Sanders on micrometer-sized colloidal systems (15). However, HS models do not provide a satisfactory description of experiments, as clear from the fact that crystals isomorphic to MgZn_{2}, CaCu_{5} (1), body centered cube AB6 (bcc-AB_{6}) (4) [also known as Cs_{6}C_{60} (7)], or quasi-crystals (16), just to name a few, have not been reported as equilibrium phases for HS (17, 18). It has also been observed that different binary systems with the same NP hydrodynamic radius (with different hydrocarbon chain length, for example) do not exhibit the same equilibrium phase (19), clearly pointing to a phase diagram that depends on other parameters besides the ratio of the two NPs diameters, which completely determines the phase diagram of the HS system (15).

The interaction between two NPs is far more complex than a HS because the polymer shell (consisting of grafted hydrocarbons or DNA) is flexible. In the limit where the grafted polymers are infinitely long (f-star limit) such potential is known analytically (20) and does reveal a very soft tail. In SE evaporated systems, however, the grafted hydrocarbon chains contain between 9 and 18 hydrocarbons (3), which are too short to be described by the f-star limit.

Motivated by the partial success of HS models, namely the imperfect but clear correlation between experimental equilibrium structures and those with high packing fraction (PF) and the need, for the reasons exposed (see also ref. 19), to consider a soft potential, we examine particles of different diameters interacting with an inverse power law (IPL):*p* the model is continuous (in *r*) and thus provides a generic example of a soft interaction; Fig. 1. In this paper, only the

IPL potentials with *p* in the range 3–12 have been used in models of soft colloids before, such as, for example, in characterizing the phase diagram of microgel particles (see review in ref. 21), thus showing that IPL potentials provide a generic description of soft particles, although, admittedly, more refined potentials are needed for more rigorous quantitative studies (22).

## Thermodynamic Parameters

The potential Eq. **1** is parameterized by

It should be noted, however, that, without loss of generality it can be set that *Discussion* below and plot of the respective PF in *SI Appendix*, Fig. S1) and being experimentally relevant. All lattices are parameterized in terms of an equivalent binary HS model **2**, but test calculations showed that allowing (

The fact that the underlying HS radii parameterizing the lattice match so well with the two radii defined by the potential Eq. **2**, as well as the strong correlation between stability of the phase and HS PF, shows that the parameter *γ* has a clear physical interpretation as the ratio of the two soft-particle radii.

## Results

The general phase diagram is a function of four coordinates, defined in Eq. **4**: two that parameterize the interaction (Eq. **2**) and two purely thermodynamic variables (Eq. **3**). The excess free energy per particle in units of **3**) can be written as (25)**11** has been extensively discussed in ref. 25. Concrete values for these coefficients are given in *SI Appendix*, Tables S2 and S3). The symbol *SI Appendix*, section SII, Eq. **S14**). The anharmonic function *Materials and Methods* as well as in ref. 25, the error introduced by this approximation is less than 0.1

The equation of state relates **5** and **6** as

The calculation of a phase diagram at a given value for the potential parameters Eq. **2** proceeds by first establishing the stable phases, determined by the condition that all eigenvalues of the dynamical matrix (DM) except for the three acoustic modes, are positive. Then, the *SI Appendix*, Tables S2 and S3), and with it, the chemical potential Eq. **7**. The stability range for each phase (for the phase diagrams reported in this study) is provided in *SI Appendix*, Table S1 for all cases considered; a given lattice is not always stable, but if it is, it is always around a finite interval around its highest PF (*η*) and becomes unstable around

The next step consists of establishing whether the stable pure *SI Appendix*, section S1):

Two illustrative examples are provided in Fig. 2. The first consists of _{2}, and NaZn_{13}. The second consists of _{2}, CaCu_{5}, and NaZn_{13}. Both pure A and B phases are fcc. The resulting phase diagram is shown in Fig. 3: In the first case all stable phases are equilibrium at least within a limited pressure range. In the second case the CsCl is not an equilibrium phase and CaCu_{5} is only stable over a very limited pressure range. In both cases, the phase diagram is different from the one obtained by direct application of the large **8**.

## Discussion

In this way, by performing the calculations outlined in the previous section, the phase diagram for all *γ*-values at 12 different values of *SI Appendix*, Figs. S2–S13) are shown in Fig. 4. The interval where each phase is not just stable, but an equilibrium phase is displayed, with its corresponding PF highlighted. At a given *γ*, many phases can exist, either because they occur with different stoichiometry (such as NaCl and AlB_{2} in Fig. 3) or because there is a phase transition at a given _{13} phase). If this latter case occurs, some loss of information results from the plots, because a full phase diagram like the one in Fig. 3 cannot be fully reconstructed from those plots as the critical pressure *γ*) is not provided.

Rather interestingly, phase diagrams for _{2} as the only nontrivial equilibrium phase. Nontrivial phases are found for

Fig. 5 summarizes all phase diagrams. In this plot, the regions in *γ*-values are shown.

## Implications for NPS

In addition to the diameter ratio *γ* that fully characterizes the phase diagram of HS models, IPL potentials have one additional parameter

### DNA Systems.

NPS in DNA systems are driven by hybridizations between A and B particles. Such hybridizations can be described by the interaction energy between A–B particles

As is clear from Fig. 5, those systems display the CsCl, AlB_{2}, bcc-AB_{6} (known as Cs_{6}C_{60} in this context) found in experiments (7, 27). There are four additional equilibrium phases in this region: NaCl, AuCu, AuCu_{3} and NaZn_{13}. Explicit calculations (7) (see also ref. 28) have ruled them out as they do not optimally hybridize the DNA shells, but those calculations do not consider the role of entropy and the possibility of coexistence. As the results presented here reveal, they could become equilibrium phases, at either small *γ* (NaCl), _{3}), or when there is an abundance of B particles, that is, _{13}). Another possibility to target these phases is to modify the DNA shell to include additional AB repulsion forces that compete against hybridizations (28⇓–30), which is achieved by inclusion of neutral (those that do not form hybridizations) DNA strands into the NP shell (28, 30). In those systems, AuCu phases have been found (28, 30).

### SE Systems.

Experimentally, the NP radius is defined as half the smallest NP–NP in a 2D hexagonal lattice of same NPs (2). The PF broadly used to interpret experimental results corresponds to the value *γ* obtained by the ratio of the smallest to the largest radii obtained this way. Here, I introduce the softness asymmetry (SA) parameter as

The plots in Fig. 6 reveal a clear correlation both vertically (PF) and horizontally (SA). In fact, such correlation is so remarkable that rescaling the SA by a factor _{2} is predicted to be the equilibrium phase over the competing MgZn_{2}, which is the one experimentally reported. The calculations show a preponderance of CsCl over the AuCu competing phase, and finally, the CaB_{6} are not found to be equilibrium phases (and are stable over a very limited PF).

The difference in free energy between MgZn_{2} and MgCu_{2}, as clear from the *SI Appendix*, Tables S2 and S3), is extremely small (but always favorable to the latter). The reasons for the preponderance of CsCl against AuCu seem less clear, although for the values of *γ* reported, both phases are structurally very close (they become identical for _{6} is stable, the free-energy difference with the bcc-AB_{6} is very small.

A closer analysis of Fig. 6 shows that the results of ref. 19 explore a wide range of the parameter _{4}C, and NaZn_{13} phases. The effective thickness used for the PbSe ligand was taken as 0.85 nm (ref. 2, table 1). The results of ref. 4, 14 nm Fe_{3}O_{4} (oleic acid), 4.6 nm Au (dodecanethiol) with the effective thickness of oleic acid estimated as 0.9 nm (by analogy with *γ*-Fe_{2}O_{3}, also from ref. 2, table 1), are also outside the range of the theoretical result. In both cases, however, should a longer effective thickness for oleic acid be used, the results would be within consistency. In any case, given the simplicity of the model, the overall agreement with the theoretical phase diagram is remarkable.

## Conclusions

The phase diagram for IPL potentials has been described and provides a paradigm for any general short-ranged soft potential; introduction of soft repulsive potential allows a characterization of a significant morphological diversity in self-assembled superlattices. In the region where phase separation into A + B can be overcome, the phase diagram is much richer than the one reported for HS (15, 17, 18). It remains an open question as to whether more general HS models (where, for example, the additivity of the radius is dropped) can account for these additional phases. More importantly, the study provides a semiquantitative description of the equilibrium crystals observed in NPS experimental systems by considering two parameters only: PF and the SA; Fig. 6. How robust predictions based on SA are is a subject for further exploration.

This paper will be followed by further studies where more sophisticated potentials or free energies are considered. There have been many important developments relating the relation of potentials to given lattice structures (32⇓–34) or characterization of optimal PFs (35⇓–37) that will definitely synergize with this work.

It is my contention that the presented calculations capture the conceptual relevant aspects of the phase diagram with soft interactions, thus providing at least a qualitative description of mesoscale NPS. Furthermore, the methods presented are completely general and can accommodate any continuous potential. In addition, this paper highlights the clearly unappreciated value of DLT: Calculation of these phase diagrams by standard thermodynamic integrations would have required enormous computational resources.

## Materials and Methods

DLT (38) provides an exact low-temperature expansion to investigate the thermodynamic properties of any phase with long-range order (23, 24). Quantitative comparisons of excess free energies in soft potentials have demonstrated agreement within less than 0.1

where *D* is the DM, and *s*-lattice basis at the unit cell ** n** along the

Both the lattice sums and the DM were calculated using the HOODLT software, which is described elsewhere (25). The calculations were run in a CPU cluster using Message Passing Interface (MPI) as implemented in the Python package PyPar. A calculation of a phase diagram at all γ for given values of

## Acknowledgments

A.T. acknowledges interesting discussions with M. Boles, C. Calero, N. Horst, C. Knorowski, O. Gang, G. Miller, D. Talapin, and D. Vaknin. I thank T. Kennedy for many important discussions and encouragement during the course of this work. Warm thanks for the use of computer cluster Cystorm at Iowa State, which has been purchased from multiple National Science Foundation (NSF) grants. This work is supported by the US Department of Energy (DOE), Basic Energy Sciences, Materials Science and Engineering Division. The research was performed at the Ames Laboratory, which is operated for the US DOE by Iowa State University under Contract DE-AC02-07CH11358.

## Footnotes

- ↵
^{1}Email: trvsst{at}ameslab.gov.

Author contributions: A.T. designed research, performed research, analyzed data, and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504677112/-/DCSupplemental.

## References

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Macfarlane RJ, et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Srinivasan B, et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Jain A,
- Errington JR,
- Truskett TM

- ↵
- ↵
- ↵
- ↵.
- Born M,
- Huang K

## Citation Manager Formats

## Sign up for Article Alerts

## Jump to section

## You May Also be Interested in

_{2}conditions, by the late 21st century, increasing temperatures could lead to reduced snowpack, drier summers, and increased fire risk, independent of changes in winter precipitation.

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.