Amorphous-amorphous transitions in granular packings

Abstract

Polyamorphs are the amorphous counterparts of the distinct crystalline phases that are found in the phase diagram of a substance. For a given system, polyamorphs have different local structures, resulting in distinct macroscopic properties, making them important for fundamental science and applications. While previous studies have mainly focused on polyamorphism occurring in systems with an open network structure, little is known how in systems of non-spherical particles the properties of polyamorphs are affected by the orientational degrees of freedom of the particles. Here we employ computational X-ray tomography to investigate how the structure of granular packs composed of spheres, icosahedra, and dodecahedra depends on their packing density. Characterization of the structure in 3D by means of a novel four-point correlation function demonstrates that these systems show a pronounced growth of the local packing anisotropy when the packing fraction is increased, inducing the emergence of two distinct polyamorphs in systems of non-spherical particles. These two structures differ in the orientational order on intermediate distances, revealing a novel class of polyamorph. The identification of these polyamorphs advances our understanding of the jamming and glass transition and will help the development of new types of functional glasses.

The medium range order in amorphous materials has a profound impact on their properties. Yuan et al. investigate how non-spherical particle shape affects granular packing and reveal amorphous-amorphous transitions associated with an orientational order of particles at intermediate length scales.

Introduction

Knowledge of the structure of liquids and glasses allows not only to predict their thermodynamic and mechanical properties¹, but also to test theories of the glass-transition which are based on thermodynamic approaches, i.e., that assume that glassy dynamics is related to a growing structural length scale^2–5. For the important case of hard-spheres (HS), experiments, simulations, and theory have provided detailed insights into the microscopic arrangement of particles in the liquid and glassy state, demonstrating that theory is able to predict with good accuracy the features of two-point correlation functions, such as the radial distribution function g(r)^1,6–8. However, recent studies have revealed that disordered systems have also highly non-trivial structural features on intermediate length scales which are not detectable in g(r), such as a pronounced anisotropic local arrangement of the particles in three dimensions or long-range correlations between coordination numbers^9–12, indicating that even for the case of HS our understanding of the structure on intermediate and large length scales is incomplete. Since recent studies have shown that many macroscopic properties depend on the medium range order (MRO) of the system, i.e., 3-8 particle diameters^13–18, this missing insight calls to investigate this MRO in more detail. However, evidence for the existence of these relationships has so far only been given in a qualitative and often indirect manner, since until recently there was no operational framework that allows to quantify the MRO. Also the identification and characterization of polyamorphs is affected by this problem, since usually the structures defining the various polyamorphs differ not in the particle arrangement within the nearest neighbor shell, but on larger distances, i.e., polyamorphs are related to the MRO^19–31. Thus probing in a quantitative manner the MRO permits to identify the presence of polyamorphs and hence advance our understanding of the nature of glass-forming systems, opening the door for the design of amorphous materials with novel properties.

Most glass-formers are composed of non-spherical particles (e.g., molecular liquids or granular systems), since the presence of the orientational degrees of freedom permits to have a larger number of different local structures, which hinders crystallization^32,33. Although various studies have probed how in such systems quantities like the total packing fraction or the nearest neighbor structure depend on the shape of the particles^32–48, much less is known about the arrangement of the particles on intermediate distances. In particular, the question on the nature of the MRO, the existence of polyamorphs, and the relation of these quantities to the shape of the particles and their orientational degrees of freedom has so far not been elucidated.

To advance on these questions, we have used computational X-ray tomography to determine the 3D structure of granular packs formed by spheres, icosahedra, and dodecahedra as a function of the packing fraction ϕ. The analysis of the structure by means of a four-point correlation function reveals that the systems with non-spherical particles show a marked change in their MRO when ϕ is increased and that this amorphous-amorphous transition is due to the orientational degrees of freedom of the particles, leading to the existence of different polyamorphs.

Results

Tapping experiment

We investigate three types of monodisperse particles: Spheres (SPHE), regular icosahedra (ICOS), and regular dodecahedra (DODE). Particles were produced with a 3D printer (ProJet MJP 2500 Plus, 0.032 mm resolution) using a plastic material (VisiJet M2R-WT, ρ = 1.12 × 10³ kg m⁻³) and have an inner diameter of d = 4 mm. More details on the particles are given in the Methods and Supplementary Fig. 1. The particles are poured into a cylindrical container (inner diameter is 150 mm) up to a height of  ≈ 140 mm, see Fig. 1a. The granular packs are driven using a mechanical shaker (DFT9100, 1000 N excitation force) with tap intensities ranging from Γ = 5g to 26g, where g is the gravitational acceleration constant. Depending on Γ, the system achieves its steady state packing fraction ϕ within 10 to 10⁵ taps, see Supplementary Fig. 3. Each tap cycle consists of a 200 ms pulse, followed by a 1.5 s interval during which the system is allowed to settle. Once the pack has reached its steady state, its structure is determined by X-ray tomography scans using a medical CT scanner (UEG Medical Group Ltd.) and image processing procedures outlined in previous studies^10,49, allowing to determine the particle positions with an accuracy better than 10⁻² d. More details on the experimental procedure are given in the Methods. Results are obtained by averaging 5 independent samples, each containing between 20,000 and 25,000 particles, after having excluded particles within a distance of 3 d from the container boundary and the free surface.

Fig. 1. Experimental setup and Γ-dependence of ϕ indicating the presence of an amorphous-amorphous transition.

a Schematic of the experimental setup, showing the cylindrical container filled with particles (purple), along with the X-ray source and detector used to determine the sample’s 3D structure. b Packing fraction ϕ as a function of tap intensity Γ for the SPHE, ICOS, and DODE systems. Dashed lines show the value ϕ_c at which the amorphous-amorphous transition takes place. Error bars represent the standard error of the mean (SEM).

The two branches of ϕ(Γ)

Figure 1b shows the Γ-dependence of the packing fraction ϕ for the three systems. As expected, ϕ increases with decreasing Γ and at the lowest amplitudes it starts to flatten out because the systems enter the glass regime, i.e., one no longer probes the steady state behavior. The curves for ICOS and DODE display a marked kink, indicating a transition between two different structures at a density ϕ_c. A weak kink is also seen in the SPHE system, although the fact that it is very close to the glass state makes it hard to detect. (Supplementary Fig. 6 shows that none of these systems crystallize, thus this is not the reason for the observed kinks.) For the case of hard spheres, a similar crossover has been documented by Philippe and Bideau⁵⁰ who speculated that it is due to the presence of two different driving mechanisms (slow relaxation vs. shock waves), but no microscopic data was provided to support this view. By monitoring the Γ − dependence of ϕ in different parts of the samples, we find that ϕ_c is independent of the location, see Supplementary Fig. 7. Since different locations have different densities and driving forces, one concludes that the kink is not a dynamical effect but instead related to a change in structure, i.e., that the systems undergo an amorphous-amorphous transition. In the following, we will denote the two ϕ(Γ)-branches that meet at ϕ_c as high-density packing (HDP) and low-density packing (LDP). Such amorphous-amorphous transitions, which indicate the presence of polyamorphism, have been documented for other systems, such as open network-forming liquids, notably water, in which structural changes are related to the presence of two different local length scales^19–25, or for metallic glasses, for which the history of the sample can give rise to different polyamorphs^26–30. However, as documented below, in our case the transition is triggered by the formation of an orientational order in the structure on intermediate distances, thus a mechanism that is very different from the one discussed in the literature. We will show that this type of order is strongly enhanced when the particles are non-spherical, thus making the transition very pronounced in the ICOS and DODE systems. Note that on the LDP branch of the DODE system, the decay of ϕ(Γ) is faster than the one for the other two systems. This is likely due to the fact that for DODE, the density of random loose packing, expected to be reached for very large Γ, is smaller than the one for SPHE and ICOS⁵¹.

Translational and orientational two-point correlation functions

In the following, we discuss how the radial pair correlation function gives insight into the anisotropy of the structure and how it is correlated with the local orientation of the particles.

The radial distribution function g(r) allows to characterize the structure of disordered materials¹, and Fig. 2a presents this correlator for the three systems at different ϕ. Since g(r) decays in an exponential manner to the limiting value of 1.0¹, we plot (g(r) − 1) multiplied by an exponential that compensates for this decay. (The g(r) are shown in Supplementary Fig. 8 and Supplementary Fig. 9.) For the SPHE system, we find that with increasing ϕ the second nearest neighbor peak starts to split, the third nearest neighbor peak starts to form a shoulder, and that the peaks at larger distances become more pronounced, in agreement with earlier findings^11,52–55. The same qualitative behavior is found for ICOS and DODE, but one observes, in particular for the DODE, also a splitting of the peaks at larger distances, i.e., beyond r ≈ 5, indicating the presence of a pronounced medium-range order. The nature of this order is clarified by considering the angular dependence of g(r) as proposed in refs. ^9,10 for the SPHE system. For this, we introduce a local coordinate system by selecting 3 particles that are mutual nearest neighbors. We define the #1 of the particles as the center of the coordinate system, the connection between #1 and #2 as the z-axis, and the plane containing the three particle as the x-z-plane, see Fig. 3a. The 3D distribution of a fourth particle around the central particle can now be measured in this local coordinate system, resulting in a four-point correlation function. In agreement with the results of ref. ¹⁰ one finds that for intermediate and large r this density distribution shows an alternating pattern with icosahedral/dodecahedral symmetry, which allows to define two complementary directions I and D that point in the direction of the vertices of these two patterns, see Inset in Fig. 2b. (In practice the g(r) is averaged over the narrow cone, opening angle is 10°, shown in the Inset.)

Fig. 2. Two-point and orientational correlation functions reveal anisotropy of the medium range order.

a g(r) for SPHE, ICOS, and DODE systems at three densities. b–d Directional correlation functions g_dir(r) and C_dir(r) for SPHE (b), ICOS (c), and DODE (d) systems. The g_dir(r) curves have been multiplied with an exponential term $\exp \left(r/\xi \right)$ to allow to see the structure at larger distances. The curves C_dir(r) have been multiplied by $6\times \exp \left(r/\xi \right)$ and shifted vertically by  − 5 (ICOS) and  − 10 (DODE).

Fig. 3. 3D particle distribution at intermediate distances exhibits dodecahedral symmetry.

a Definition of the local coordinate system used to calculate the density distribution around a central particle #1. b Three-dimensional particle distribution of the three systems at a low and high density and for three different distances r, corresponding approximately to the location of the 3rd, 4th, and 5th peak in g(r). The width of the shell to determine the distribution has a thickness 0.1d. The same color bar range, quantified on the right, is used for each row.

For the case of the non-spherical particles, one can define the local coordinate system directly from the particles via the line that connects the center of a particle to the center of one of its faces ("fac direction”) or to one of its vertices ("ver direction”). (In practice one uses also here a narrow cone, see Insets of Fig. 2c and d and Supplementary Fig. 10 and Supplementary Fig. 11). This decomposition of g(r) into g_dir(r) (dir = I, D, ver, fac) is presented in Fig. 2(b)–(d) and one recognizes that the mentioned splitting of the peaks is due to the anisotropy of g(r) in that the ver and D directions have marked peaks with a uniform shape while the peaks in the fac and I directions are located between the ones for ver/D, but have a shape that depends strongly on r, in particular for the ICOS and DODE systems. (For the SPHE system, see also ref. ¹¹.) Hence, one concludes that the MRO depends significantly on the shape of the particle and that this dependence is mainly due to the way the order is formed in the fac/I direction.

For polyhedral particles like ICOS and DODE, one can also define an orientational correlation function C(r) which measures the angle between the line connecting particles j and i and the normal vector of the face on particle j that is crossed by this line, n_j:

$$C\left(r\right)=\frac{1}{N}{\sum }_{i,j=1}^N⟨\arccos \left(\frac{{\mathbf{r}}_j-{\mathbf{r}}_i}{\mid {\mathbf{r}}_j-{\mathbf{r}}_i\mid }\cdot {\mathbf{n}}_j\right)\delta \left(r-\mid {\mathbf{r}}_j-{\mathbf{r}}_i\mid \right)⟩-C_{\mathrm{r}}.$$ 1

Here, r_i is the position of particle i and C_r is the expectation value of the mentioned angle at large distances, which is given by 0.3147 and 0.3896 for the ICOS and DODE systems, respectively, see Methods. Similar to g(r), this correlator can again be decomposed into its directional components, C_dir(r), which are included in Fig. 2c and d as well. (Since also these functions decrease in an exponential manner with r, we have multiplied them by an exponential function that compensates this decay. The ϕ-dependence of the decay length is discussed below.) One notes that C_fac(r) is anti-correlated with g_fac(r) when r/d is an integer, indicating that in the face direction, high positional order goes with strong orientational alignment, i.e., a small value of C. Conversely, the C_ver(r) curves track the g_ver(r) curves, suggesting that in this direction, pronounced positional order correlates with weak orientational alignment. We thus conclude that in the face direction, positional as well as orientational orders are high, resulting in well-ordered packing. In contrast, in the vertex direction, positional order remains relatively strong, while the orientational order is weak.

Our analysis of the orientational-dependent g(r) shows that the positional correlation function is highly anisotropic and, depending on the direction considered, is correlated/anti-correlated with the local orientational order of the particles. In the following, we will thus probe how this interplay between translational and orientational order affects the MRO as a function of ϕ and make a connection with the kink seen in Fig. 1, hence show the existence of polyamorphism.

Multi-point correlation function

To elucidate the 3D structure of the systems, we present in the following a recently introduced method that allows to characterize this structure in a quantitative manner.

The seemingly simplest way to identify the mechanism that causes the kink in ϕ(Γ) is to probe how the location of the peaks in g(r) or g_dir(r) depends on ϕ, since it can be expected that these positions are directly related to ϕ. However, it is found that none of these quantities shows at ϕ_c a significant change in their ϕ-dependence (see Supplementary Fig. 12), i.e., simple two-point correlation functions are not sufficiently sensitive to detect the amorphous-amorphous transition. To characterize the three-dimensional structural properties, we employ the four-point correlation function: For each particle, we defined the local coordinate system as described above (see Fig. 3a). Using this reference frame, we measure the four-point correlation density ρ(r), i.e., the density field defined on the surface of a sphere of radius r centered at the origin of the coordinate system. More details are given in the Methods.

Figure 3b presents ρ(r) for the three systems at a low and high density (left and right columns, respectively). The distances r correspond to the location at which the density field has a pronounced dodecahedral symmetry, thus close to the location of the peaks in g(r). (See below for details on how this is quantified.) If r and ϕ are small, the density fields for the three systems have a very similar symmetry and value of the density, and for SPHE and ICOS this is also the case at high ϕ. This demonstrates that the short-range packing is basically independent of the shape of the particle. For the low packing fractions, one notes that the signal for DODE decays very quickly with r (different rows), while for SPHE this decay is slower. (ICOS shows an intermediate behavior.) This implies that the structural correlation length for DODE is significantly shorter than the ones of SPHE, a behavior that will be quantified later. For a high packing fraction the decay of the signal strength with increasing r is significantly slower, i.e., the structural correlation length has increased. Remarkably, one finds that for the DODE system, this increase is very strong since now one has a ρ(r) that, at large r, is much more structured than it is for low ϕ. This result indicates that there is a mechanism that allows the DODE system to improve quickly its packing efficiency, and hence its structural correlation length, at high ϕ.

To quantify the strength of the symmetry of the ρ(r) field, we decompose it into spherical harmonics $Y_l^m\left(\theta ,\phi ,r\right)$ with angular variables θ and φ: $\rho \left(\theta ,\phi ,r\right)={\sum }_{l=0}^{\infty }{\sum }_{m=-l}^l{\rho }_l^m\left(r\right)Y_l^m\left(\theta ,\phi \right)$, where the expansion coefficients, ${\rho }_l^m$, are provided in the Methods. Since the local structure around a particle is close to an icosahedron, we focus in the following l = 6^9,10. The square root of the angular power spectrum, $S_{\rho }\left(l,r\right)={\left[{\left(2l+1\right)}^{-1}{\sum }_{m=-l}^l\mid {\rho }_l^m\left(r\right){\mid }^2\right]}^{1/2}$, characterizes the anisotropy of the density distribution, and shows an oscillatory r-dependence with an envelope that decays in an exponential manner, see Fig. 4a–c, in agreement with the results from refs. ^9,10. These curves also reveal that the height of the peaks, and thus the prominence of the positional order, depends not only on ϕ but also on the position of the peak, i.e., on the length scale considered.

Fig. 4. S_ρ(l = 6, r) characterizes the strength of the anisotropy in the density distribution.

S_ρ(l = 6, r) is shown for the (a) SPHE, (b) ICOS, and (c) DODE systems. At large distances, the local maxima decrease exponentially with r. d–f The height of the first six peaks of S_ρ as a function of ϕ for the SPHE, ICOS, and DODE systems. These heights are normalized by $S_{\rho }\left({\varphi }_{\min }\right)$, where ${\varphi }_{\min }$ is the lowest packing fraction considered. Vertical blue bars indicate ϕ_c, the location of the amorphous-amorphous transition observed in Fig. 1, and demonstrate that the ϕ-dependence of the peak heights show a marked change at ϕ_c. Error bars represent the standard error of the mean (SEM).

To quantify this ϕ and r-dependence, we present in Fig. 4d–f the height of the peaks in S_ρ(6, r) as a function of ϕ and one recognizes that at small ϕ the nearest neighbor peak increases steadily, reflecting the ordering of the structure on this length scale upon compression. For the SPHE system, one finds also a strong increase of the order on larger length scales, while for the ICOS and DODE systems, the growth of the peaks at larger distances is only modest. Before the packing fraction reaches ϕ_c, vertical bars in the graphs, the growth of the peak height slows down and the curves saturate, indicating that the short-range packing has reached a (quasi) optimal order. For ϕ ≳ ϕ_c, all systems show a new rapid growth of the peak height, signaling the emergence of a new structural order. Note that this accelerated growth is most prominent at larger distances, i.e., it is the MRO that is changing. Instead, the ϕ-dependence of the first peak is very mild and shows no particular feature at ϕ_c, which explains why previous studies that focused mainly on the short-range order have not detected the structural transition that occurs at ϕ_c.

Our quantitative analysis of the 3D structure shows that with increasing ϕ this structure becomes more pronounced, and that the ϕ-dependence changes at ϕ_c. For the SPHE system, this break is observed only at relatively large distances, while for the ICOS and DODE systems, it is seen already at intermediate distances, which explains why in Fig. 1b the kink is not very pronounced for the SPHE system. Note that S_ρ(l, r) is independent of the orientation of the particles, i.e., it is a purely positional observable. It is therefore important to probe how the translational and orientational degrees of freedom are coupled to each other, which is done in the next section.

Correlation length scales

In this section, we introduce several length scales that allow to quantify how the extent of the MRO changes as a function of packing fraction.

The exponential decay of S_ρ(r), presented in Fig. 4a–c, can be used to define a correlation length ξ_S, and this can also be done for g(r) and C(r), thus defining ξ_g and ξ_C, see Supplementary Figs. 13, 14 and 15. The ϕ-dependence of these quantities are displayed in Fig. 5a and one recognizes that for the SPHE system ξ_g and ξ_S track each other and are basically constant, demonstrating that for this system, the nature of the MRO is not changing in a significant manner. In contrast to this, we find that for the ICOS and DODE systems, the scales ξ_S and ξ_C are constant for ϕ ≲ ϕ_c, but start to increase quickly beyond this packing fraction. (The g(r) for ICOS and DODE systems decay in a complex manner, see Supplementary Fig. 9, and hence cannot be used to obtain ξ_g). Thus, one concludes that close to ϕ_c the MRO of the ICOS and DODE systems starts to become more pronounced and that this change affects the translational and orientational order in the same manner.

Fig. 5. Length scales reveal the two amorphous structures.

Panels (a) and (b) show the length scales ξ and L, respectively, as functions of ϕ for the SPHE, ICOS, and DODE systems. The threshold used to define L for S_ρ is 10⁻³. For g(r) in the SPHE system and C(r) in the ICOS and DODE systems, the thresholds are 1.5 × 10⁻², 1.3 × 10⁻³, and 2.3 × 10⁻³, respectively. Error bars represent the standard error of the mean (SEM).

That for the SPHE system the length scale ξ_g is independent of ϕ might surprise, since with increasing ϕ the structure is expected to change. We emphasize, however, that ξ is characterizing only the decay length, while the details of the structure affect also the prefactor of the exponential. To take into account the combined effect of the prefactor and ξ, we define a new length scale L_α by requiring that the correlation function α ∈ {g(r), S_ρ(r), C(r)} has decayed to a fixed threshold. Figure 5b demonstrates that for the SPHE system, the L_g and L_S do indeed increase in the whole range of ϕ, in line with the data presented in Fig. 4(d). This demonstrates that for the SPHE system, it is the short-range order that determines the order on intermediate length scales.

For the ICOS and DODE systems, the ϕ-dependence is different: While for ϕ ≲ ϕ_c, L_S grows only weakly and flattens when approaching ϕ_c, L_C increases quickly. This shows that at low ϕ the densification is a combination of local orientational alignment of the particle and, to a lesser extent (see Fig. 4e, f), an increasing positional order. Note that at these values of ϕ, L_C is smaller than L_S, demonstrating that the orientational order is more short ranged (less important) than the positional one, making that the structure of the MRO is qualitatively similar to the one of the SPHE system (although decaying a bit faster, panel (a)). At ϕ_c the scale L_C reaches L_S and the presence of this orientational order on that length scale triggers an accelerated growth of the positional order since the particles can better pack due to the decreasing orientational disorder, in agreement with Fig. 4d–f.

The length scales that grow quickly for packing fractions above ϕ_c hint the existence of a structure formed by particles that are correlated. To probe the properties of this structure, we use the tetrahedral order parameter δ which measures how a tetrahedron formed by 4 nearest neighbor particles deviates from a perfect tetrahedron: $\delta =e_{\max }/d-1$, where $e_{\max }$ is the length of the longest edge of the tetrahedron⁵⁶. Defining as “regular” the tetrahedra that have a δ smaller than a threshold δ_c, one can use a nearest neighbor criterion to identify clusters that are formed by these regular tetrahedra, i.e., that have a strong structural correlation⁷. The radius of gyration of cluster k is then given as $R_g^2\left(k\right)=\frac{1}{2}{\sum }_{i,j}^{N_c\left(k\right)}\mid {\mathbf{r}}_i-{\mathbf{r}}_j{\mid }^2$, where the double sum runs over all N_c(k) particles in the cluster. One now defines the mean radius of gyration of these clusters as $R_c^2=\frac{2{\sum }_k{R}_g\left(k\right)N_c^2\left(k\right)}{{\sum }_k{N}_c^2\left(k\right)}$. In Fig. 6, we show the ϕ-dependence of R_c and one recognizes that this size shows a marked increase of growth at ϕ_c, demonstrating that at this packing fraction the structure of connected regular tetrahedra starts to expand quickly, confirming the results on the length scales shown in Fig. 5 that indicate a new form of order beyond ϕ_c. Note that the length scale R_c has a rather different nature than the one of ξ and L: The former one describes the typical size of the clusters formed by regular tetrahedra, but this quantity is not sensitive to the shape of such clusters. Instead, ξ and L probe directly the decay of correlation for the positional and orientational order as a function of the distance from a particle.

Fig. 6. Size of cluster formed by regular tetrahedra.

The thresholds δ_c to define regular tetrahedra are 0.245, 0.299, and 0.300 for the SPHE, ICOS, and DODE systems, respectively. A mild change of these values does not affect the presence and the location of the kinks seen in the curves. Error bars represent the standard error of the mean (SEM).

Although the length scales that we have considered highlight different aspects of the 3D structure of the pack (position, orientation, short-range order, MRO), all of them show a clear kink at ϕ_c, demonstrating that this is a robust feature and that the various observables are strongly coupled to each other. Therefore one concludes that the kink in ϕ(Γ), Fig. 1b, is directly related to the change of the ϕ-dependence of these length scales, i.e., a qualitative change of the structural order due to the fact that the length scale of the orientational order has caught up with the one of the positional order, triggering the formation of an improved local packing which corresponds to a new amorphous state.

Discussion

Previous studies have demonstrated that the local packing of HS becomes increasingly anisotropic if ϕ is increased, and features particle density shells with a symmetry that alternates between icosahedral and dodecahedral^10,12. At low and intermediate ϕ, this type of positional anisotropy is also present in the local structure formed by our non-spherical particles, as can be concluded from the similarity of the density field shown in Fig. 3b. This demonstrates that such a particle arrangement can be expected to be generic if the shape of the particles does not deviate too much from spherical symmetry. In contrast to this universal behavior, our experiments reveal that, at large ϕ, non-spherical particles can pack in a novel kind of structure that displays an enhanced orientational ordering of the particles on intermediate length scale, see the schematics in Fig. 7. (Supplementary Fig. 20 shows that this increased positional medium range order is most pronounced in the direction of the faces.) This orientational transition between two polyamorphs trades high orientational entropy versus increased packing efficiency (thus high positional entropy). This transition bears resemblance to the displacive transition observed in crystalline systems and the orientational transitions occurring in plastic crystals^57,58, since also in these cases the phase transition is related to a small change in the local structure. The occurrence of this transition rationalizes the existence of the two branches in Fig. 1b. We emphasize that this amorphous-amorphous transition at ϕ_c is an equilibrium phenomenon, i.e., neither of the two steady states is affected by kinetic effects, as it is often the case in the amorphous-amorphous transitions found in metallic glasses^26–30. Instead, the transition is completely reversible if the driving amplitude Γ crosses a critical threshold.

Fig. 7. Schematic representation of polyamorph structures.

Panels (a) and (b) show the low- and high-density phases, respectively. Green: The central particle. Red: Particles that are in the direction of the faces of the central particle. The sectors that have been used to calculate the directional g(r) for the faces, see Fig. 2, are colored in light red. Blue: Particles in the directions of the vertices of the central particles. The sectors that have been used to calculate the directional g(r) for the vertices are colored in light blue. The arrows represent the vector n_j characterizing the alignment of the faces of the particle with the faces of the central particle (see text for details). In the LDP, the particles have only a moderate orientational order, while in the HDP, this order becomes pronounced and triggers an increased positional order in the directions of the faces. This ordering allows for a tighter packing of the particles.

It is important to point out that the amorphous-amorphous transition documented in the present work is not sharp, i.e., none of the order parameters (or their derivatives) shows a singular behavior. Instead, similar to the case of amorphous-amorphous transitions seen in atomic systems, the transition should be seen as a rapid crossover between two amorphous states that have a different structure. It can, however, be expected that by improving the statistics (more samples, larger system sizes, and state points closer to ϕ_c), the sharpness of the transition can be enhanced. Such studies are important and interesting since they should allow to probe the transition in more detail. This type of study is thus promising work for the future.

As the particle shape approaches a sphere, the transition is expected to become less pronounced, as shown in the Supplementary Information with data (Supplementary Fig. 21) for one additional particle shape. Also, the introduction of spherical particles in a one-component system of non-spherical particles is likely to influence the transition and might wash it out at sufficiently high concentrations. Further studies on the minimal anisotropy required to observe the amorphous-amorphous transition are an interesting topic for future investigations.

We note that variations in properties of the particle surface, such as the friction constant or the surface roughness, have been shown to influence both packing fraction and coordination number in granular systems⁵⁹. Although such effects are likely to affect the value of transition packing fraction ϕ_c, they are not expected to alter the mechanism giving rise to the amorphous-amorphous transition, i.e., this phenomenon is a robust feature of granular systems of particles that are noticeably non-spherical. Also differences in driving (e.g., tapping versus vibration) can modify the efficiency of energy injection⁶⁰, and may shift the apparent ϕ_c (or the location of the kink in ϕ(Γ)), but do not fundamentally change the amorphous-amorphous transition itself since this transition is due to the interplay between translational and orientational degrees of freedom, i.e., a competition that is independent of the driving protocol.

Finally, we mention that the presence of two different branches in ϕ(Γ) can be expected to influence the dynamic properties of the system. Such a change in the dynamics has indeed been found in thermal glass-formers that show a fragile to strong transition when the local structure undergoes a change of its symmetry²⁵. Evidence that this is indeed the case also in our systems can be inferred from Supplementary Fig. 3 in which one sees, in particular for the DODE system, that at short times the curves for large Γ fall on a master curve, while for small Γ they shift continuously to the right, i.e., the dynamic response is changing qualitatively. At present, our statistics is not sufficient to probe such dynamical features in detail, but future experiments should allow to address this question.

Methods

Experiment

We consider particles of three shapes: Spheres (SPHE), regular icosahedra (ICOS), and regular dodecahedra (DODE). To determine the orientation of the particles through image processing, we created a chamfered cuboid cavity inside the particles, the center of which coincides with the center of the particle. Supplementary Fig. 1a shows a schematic of this cavity, with the lengths of its three symmetry axes being 3.2 mm, 2 mm, and 1.2 mm. Supplementary Fig. 1b, c depict sketches of the ICOS and DODE particles. Since the rotational symmetry group of the cuboid is a subgroup of that of the icosahedron/dodecahedron, the orientation of the particles can be determined by identifying the directions of the three principal axes of this cavity. Regarding potential printing imperfections, our 3D printer achieves a layer resolution of 0.032 mm and a dimensional accuracy of 0.025-0.05 mm over 25.4 mm. Since our particles have an inner diameter of 4 mm, these errors are two orders of magnitude smaller than the particle size. Furthermore, all particles of a given shape were printed in a single batch, ensuring consistent material properties and surface finish. Consequently, it can be expected that these minor surface variations do not impact the observed amorphous-amorphous transition in a significant manner. Supplementary Fig. 2 shows photographs of the particles, and one recognizes that on the length scale of the particles, their surface roughness is very small.

The CT scans have a spatial resolution of 0.2 mm, and by applying an image processing analysis similar to the one used in previous studies^10,49, the current analysis allows to obtain a high accuracy in our experimental data with estimated errors less than 10⁻²d for particle center positions and 2. 5° for particle orientations.

Between successive taps, we impose a 1.5 s waiting time to ensure that transient shaker oscillations fully decay and the system reaches mechanical rest. On the particle scale, the packing structure stabilizes within  ≲ 0.1 s, as verified by an accelerometer at the container base, a time scale which is consistent with prior work on dissipative relaxation in tapped systems⁶¹. While slow mechanical evolutions may continue for several minutes⁶², they do not affect the structural metrics reported here.

Initially, the packings are prepared in a loose state by applying 30 taps of high intensity, Γ = 22g − 26g. Subsequently, we tap the system with an intensity Γ and obtain the corresponding steady state configurations. To estimate the number of taps required to reach steady state, we track the evolution of the packing fraction ϕ as a function of tap number. Examples for the ICOS and DODE systems at different Γ values are shown in Supplementary Fig. 3a, b, respectively. A larger Γ results in a shorter transient and a lower stationary ϕ. Both ICOS and DODE systems require significantly more taps to reach a steady state compared to the SPHE system⁵⁹, since the presence of the flat faces considerably slows down the relaxation dynamics. To ensure that the system has reached the steady state, we apply between 10 and 10⁵ taps, depending on Γ.

In order to suppress crystallization of the packing during the compaction process, we have covered the inner walls of the container with half-spheres (diameters of 5 mm and 8 mm) made of Acrylonitrile Butadiene Styrene plastic. These half-spheres were randomly fixed to the inner surface of the container, establishing a permanently disordered boundary layer due to the non-uniform spacing between the glued particles. Previous work demonstrates that this approach effectively suppresses crystallization by disrupting the development of any nascent ordered structures–particularly after removal of particles located within a radial distance of 3 d from the container walls⁵⁹. Furthermore, by excluding from the structural analysis all the particles within this 3 d boundary zone, the influence of the walls on the structure can be neglected, see Supplementary Information.

Local coordinate system

For all systems, we analyze the 3D structure of the samples by introducing a local coordinate system as suggested in ref. ⁹. For this, we select any three nearest-neighbor particles (see Supplementary Fig. 10a) and we define the position of particle #1 as the origin, the direction from particle #1 to particle #2 as the z-axis, and the plane containing the three particles as the x-z plane and the y-axis is orthogonal to it. This local reference frame allows therefore to determine the 3D distribution of the particles that have a distance r from the central particle, i.e., a four-point correlation function.

For the ICOS and DODE systems, the local coordinate system can also be defined based on the single particle’s orientation determined by its shape, see Supplementary Fig. 10b and c. We first set the center of mass of the particle as the origin. For an ICOS (DODE) particle, the z-axis is defined as the direction from this center to a vertex V1 (face center F1). The x-z plane is defined as the plane containing the mass center, V1 (F1), and one of V1’s neighbor vertices, V2 (F1’s neighbor face center F2), with the y-axis orthogonal to this plane. Note that due to the symmetry of the shape, each ICOS/DODE particle possesses 60 equivalent local coordinate systems, and to get our results, we have taken the average over all of them.

Angular power spectrum for ρ(r)

To characterize the angular dependence of the particle density field ρ(r), we expand it in spherical harmonics. The expansion coefficients are given by

$${\rho }_l^m\left(r\right)={\int }_0^{2\pi }d\phi {\int }_0^{\pi }d\theta \sin \theta \rho \left(\theta ,\phi ,r\right)Y_l^{m*}\left(\theta ,\phi \right),$$ 2

where $Y_l^{m*}\left(\theta ,\phi \right)$ denotes the complex conjugate of the spherical harmonic $Y_l^m$.

In practice, rather than constructing a surface density on a grid, we treat the neighbors within a thin radial shell as a discrete point set on the unit sphere. We consider a shell centered at radius r with width Δr = 0.2 d, and define ${\widehat{\mathbf{u}}}_i=\left({\theta }_i,{\phi }_i\right)i=1,\dots ,N_r$ as the unit vectors from the central particle to all particles whose separation lies in r − Δr/2 ≤ ∥r_i∥ < r + Δr/2.

We define the angular distribution on the shell by

$$n_r\left(\widehat{\mathbf{u}}\right)=\frac{1}{N_r}{\sum }_{i=1}^{N_r}\delta \left(\widehat{\mathbf{u}}-{\widehat{\mathbf{u}}}_i\right),$$ 3

so that the spherical-harmonic coefficients are calculated directly by averaging,

$${\rho }_l^m\left(\mathbf{r}\right)={\int }_{S^2}{n}_r\left(\widehat{\mathbf{u}}\right)Y_{\ell m}^{*}\left(\widehat{\mathbf{u}}\right)d\mathrm{\Omega }=\frac{1}{N_r}{\sum }_{i=1}^{N_r}{Y}_{\ell m}^{*}\left({\theta }_i,{\phi }_i\right).$$ 4

Directional measurements

The regular shape of the ICOS/DODE particles have a high symmetry, making many orientations of these particles statistically equivalent. To characterize the translational and orientational correlations in different directions for the ICOS and DODE systems, we consider two specific directions, referred in the main text as “fac" and “ver", defined as the directions from the particle’s center of mass to the face center and vertex, respectively, see Supplementary Fig. 11. Particles located within a cone of width 10°, having its apex at the center of mass of the central particle and its axis aligned along these directions, are used to calculate the average of the observable in this direction.

For the SPHE system, the density distribution of the shell of the first nearest neighbors exhibits an icosahedral structure^9,10,63,64. By definition of the local coordinate system in the SPHE system, the directions “I” and “D” correspond to the “fac” and “ver” directions in the DODE system, respectively.

The long-range limit of the orientational correlation function

As described in the main text, ${\theta }_{ij}=\arccos \left(\frac{{\mathbf{r}}_j-{\mathbf{r}}_i}{\mid {\mathbf{r}}_j-{\mathbf{r}}_i\mid }\cdot {\mathbf{n}}_j\right)$ represents the angle measuring the alignment between two polyhedra, i and j. (The connecting line between the particles makes an angle θ_ij with the face on particle j that is crossed by this line, and n_j is the normal vector of this face.) As defined in the main text, the average of this angle, C(r) + C_r, approaches at large r a constant, and this limiting value depends on the type of polyhedron. We compute the probability distribution function of θ_ij by assuming that there is no correlation between the orientations of the two particles, i.e., the orientation of particle j is random, an assumption that is reasonable for a disordered system when r is large. The unnormalized distribution is given by the following piecewise function:

$$f\left(\theta \right)=\left\{\begin{array}{cc}2\pi \sin \theta ,\hfill & 0\le \theta \le {\alpha }_1,\hfill \\ 2n_{ef}\left[\frac{\pi }{n_{ef}}-\arccos \left(\frac{1}{2c\tan \left(\frac{\pi }{n_{ef}}\right)\tan \theta }\right)\right]\sin \theta ,\hfill & {\alpha }_1<\theta \le {\alpha }_2,\hfill \\ 0,\hfill & \theta >{\alpha }_2,\hfill \end{array}\right)$$ 5

The boundaries of the piecewise intervals, α₁ and α₂, are given by

$${\alpha }_1=\arctan \left(\frac{1}{2c\tan \left(\pi /n_{\mathrm{ef}}\right)}\right),$$ 6

$${\alpha }_2=\arctan \left(\frac{1}{2c\sin \left(\pi /n_{\mathrm{ef}}\right)}\right),$$ 7

where $c=\frac{d}{2a}$ is the ratio between the radius of the internal sphere and the edge length of the polyhedron, and n_ef is the number of edges for each face of the polyhedron. For the ICOS and DODE systems, these parameters are given by $c=\frac{1}{12}\sqrt{3}\left(3+\sqrt{5}\right)$, $c=\frac{\sqrt{250+110\sqrt{5}}}{20}$, and n_ef = 3, 5, respectively. Therefore C_r can be expressed as,

$$C_r=\frac{{\int }_0^{{\alpha }_2}\theta f\left(\theta \right)d\theta }{{\int }_0^{{\alpha }_2}f\left(\theta \right)d\theta },$$ 8

giving C_r = 0.3147 and 0.3896 for the ICOS and DODE systems, respectively.

Author contributions

H.Y., W.K. and Y.W. designed the research. H.Y. performed the experiment. H.Y., W.K. and Y.W. analyzed the data and wrote the paper.

Peer review

Peer review information

Nature Communications thanks Paolo Rissone, and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data supporting the findings of this study are available in the Supplementary Information. Due to the large size, the CT images analyzed are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing of interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-65575-5.