Uncovering the intrinsic size dependence of hydriding phase transformations in nanocrystals

Journal name:
Nature Materials
Year published:
Published online


A quantitative understanding of nanocrystal phase transformations would enable more efficient energy conversion and catalysis, but has been hindered by difficulties in directly monitoring well-characterized nanoscale systems in reactive environments. We present a new in situ luminescence-based probe enabling direct quantification of nanocrystal phase transformations, applied here to the hydriding transformation of palladium nanocrystals. Our approach reveals the intrinsic kinetics and thermodynamics of nanocrystal phase transformations, eliminating complications of substrate strain, ligand effects and external signal transducers. Clear size-dependent trends emerge in nanocrystals long accepted to be bulk-like in behaviour. Statistical mechanical simulations show these trends to be a consequence of nanoconfinement of a thermally driven, first-order phase transition: near the phase boundary, critical nuclei of the new phase are comparable in size to the nanocrystal itself. Transformation rates are then unavoidably governed by nanocrystal dimensions. Our results provide a general framework for understanding how nanoconfinement fundamentally impacts broad classes of thermally driven solid-state phase transformations relevant to hydrogen storage, catalysis, batteries and fuel cells.

At a glance


  1. Pd nanocube characterization and luminescence.
    Figure 1: Pd nanocube characterization and luminescence.

    ad, Scanning electron micrographs of Pd nanocubes of side length 14±1 nm (a), 32±2 nm (b), 65±4 nm (c) and 110±12 nm (d). e,f, Luminescence spectra of 14 nm Pd nanocubes at 22 °C decreasing during hydriding (e) and increasing during dehydriding (f). Data, for each set of conditions, are normalized by dividing luminescence values by the luminescence of the metal phase.

  2. Nanocube hysteresis loops and size-dependent thermodynamics.
    Figure 2: Nanocube hysteresis loops and size-dependent thermodynamics.

    a, Pressure–luminescence (PL) hysteresis loops at 22 °C and 82 °C obtained by hydriding and dehydriding Pd nanocubes of four sizes (bulk Pd curves shown as reference). Horizontal dotted lines show calculated coexistence pressures Pmid. Inset: percentage increase in strain calculated from full-width at half-maximum of XRD peaks before and after hydriding of nanocubes and 65 nm Pd film. Detailed calculations are shown in Supplementary Table S3, and Figs S22 and S23. b, Bulk-normalized Van’t Hoff plots for nanocubes (vertical axis quantifies the difference in Pmidbetween nanocubes and bulk samples, βΔGnorm  =  ln(Pmid/Pmidfilm)). Here βindicates the inverse temperature, 1/T. Error bars are calculated from 5 independent measurements. c, Analogous data from Ising model magnetic field–magnetization hysteresis loops: vertical axis is the difference βΔGnorm  =  2hmid/T between midpoints of hysteresis loops for nanocubes and bulk samples. Only when the nanocube surface preference ΔJ for one phase increases sufficiently strongly with temperature (upper panel) are experimental trends recovered. The choice ΔJT3is indicative only. A constant bias ΔJ (lower panel) gives a different trend, and when nanocube surfaces prefer neither phase (ΔJ  =  0, not shown) the midpoint magnetic field remains pinned to the bulk value. Comparison of b and c suggests that, experimentally, nanocube surfaces favour the hydride phase, and do so more strongly as temperature increases.

  3. Nanoconfined, thermally driven nucleation provides a natural explanation for the strong size dependence of nanocube kinetics.
    Figure 3: Nanoconfined, thermally driven nucleation provides a natural explanation for the strong size dependence of nanocube kinetics.

    a,b, Hysteresis loop widths (vertical axis is ln(Phydride/Pdehydride)) in experiment (a) vary with temperature and nanocube size in a manner similar to hysteresis loop widths in Ising model simulations (b; vertical axis is 2 (hforwardhreverse)/T). This similarity suggests that size-dependent trends observed in experiment are a natural consequence of nanoconfined, thermally driven nucleation. Experimental error bars are calculated from 5 independent measurements. c, To see this, consider Ising model magnetization–magnetic field hysteresis loops for small (red, L  =  3) and large (black, L  =  9) nanocubes (here we consider inert exteriors, ΔJ  =  0, for simplicity). Near the coexistence point h  =  0 there exists a free energy barrier to nucleation that frustrates phase change: one must wait for thermal fluctuations to surmount this barrier and generate a critical nucleus of the new phase. (N is the number of lattice sites in the nucleating cluster of the new phase). Crucially, this barrier is smaller in the smaller nanocube (see Fig. 4). As a result, phase transformation in the smaller cube occurs closer to the coexistence point h  =  0 (in both directions), and hysteresis is less pronounced than in the larger cube. Here ρ is the scaled magnetization, as defined in the Methods.

  4. Free energy barriers to phase change are controlled, near phase coexistence, by nanocube size.
    Figure 4: Free energy barriers to phase change are controlled, near phase coexistence, by nanocube size.

    a, Phase diagram for the bulk Ising model obtained using Gibbs ensemble simulations. The simulation regime explored in this Article (shaded), appropriate to our experimental regime, is far from the critical point, and so phase change near phase coexistence proceeds through nucleation. b, In a bulk system at phase coexistence, the size of the critical nucleus and free energy barrier to nucleation are infinitely large (curve labelled bulk). However, enclosing a phase transition in a finite box limits the size of the critical nucleus, rendering the free energy barrier to phase transformation finite. The smaller the cube, the smaller the free energy barrier to nucleation. Consequently, less hysteresis is observed in dynamical simulations (see Fig. 3). c, Simulation reveals that nucleation barriers in nanocubes at phase coexistence scale as c1L+c2L2 (c1 and c2 are constants), where L is cube side length. This scaling is consistent with a simple argument (Supplementary Section SIII) that assumes the critical nucleus is a two-dimensional film on an interior face of the nanocube.

  5. Nanocube kinetics resulting from sudden, large pressure changes suggest that phase change begins at nanocube surfaces.
    Figure 5: Nanocube kinetics resulting from sudden, large pressure changes suggest that phase change begins at nanocube surfaces.

    a, Luminescence data obtained for Pd nanocubes at 53 °C from experiments involving sudden pressure changes show enhanced kinetics of both hydriding and dehydriding transitions as nanocube size decreases. Data are normalized by mass of the nanocubes deposited. b, A similar hierarchy is seen in the Ising model (on sudden changes of magnetic field) whenever surface and bulk energy scales are such that phase change begins at nanocube surfaces (see snapshots (c)). c, In the snapshots, blue represents lattice cells that are in the metal phase and grey represents the periodic walls of the nanocube. If phase change begins instead in cube interiors, decreasing nanocube size leads to a slowing of kinetics (see Supplementary Fig. S16). The comparison between simulation and experiment thus suggests that both hydriding and dehydriding transformations begin at cube surfaces.


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Author information

  1. These authors contributed equally to this work

    • Rizia Bardhan &
    • Lester O. Hedges


  1. Molecular Foundry, Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

    • Rizia Bardhan,
    • Lester O. Hedges,
    • Stephen Whitelam &
    • Jeffrey J. Urban
  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, USA

    • Cary L. Pint &
    • Ali Javey
  3. Berkeley Sensor and Actuator Center, University of California, Berkeley, California 94720, USA

    • Cary L. Pint &
    • Ali Javey
  4. Present addresses: Department of Chemical and Biomolecular Engineering, Vanderbilt University, Nashville, Tennessee 37235, USA (R.B.); Department of Mechanical Engineering, Vanderbilt University, Nashville, Tennessee 37235, USA (C.L.P.)

    • Rizia Bardhan &
    • Cary L. Pint


J.J.U. and R.B. conceived and designed the experiments. R.B. synthesized and characterized the nanoparticles and performed the experiments. C.L.P. designed and built the optical cell; C.L.P. and R.B. built the high-vacuum gas-flow system. L.O.H. and S.W. conceived the simulation protocol, and L.O.H. carried out the simulations. R.B., L.O.H., S.W. and J.J.U. analysed the data and wrote the manuscript. A.J., S.W. and J.J.U. discussed the results and commented on the manuscript.

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