Alexandre Dupas
email: dupas@sic.univ-poitiers.fr

Guillaume Damiand
email: damiand@labri.fr

Comparison of Local and Global Region Merging in the Topological Map ⋆ ⋆⋆

Keywords: Combinatorial maps, Intervoxel boundaries, Region merging, Image segmentation

The topological map is a model that represents 2D and 3D images subdivision. It aims to allow the use of topological and geometrical features of the subdivision in image processing operations. When handling regions in an image, one of the main operation is the region merging, for example in segmentation process. This paper presents two algorithms of region merging in 3D topological maps: one local which modifies locally the map around merged regions, and another one global which runs through all the elements of the map. We study their complexities and present experimental results to compare both approaches.

Introduction

Region based segmentation consists in partitioning an image into connected sets of pixels or voxels called regions. One of the approaches, derived from the splitand-merge [START_REF] Horowitz | Picture segmentation by a directed split and merge procedure[END_REF] methods and called bottom-up, begins with an over-partition of the image, and decreases the number of regions using successive region merging operations. These algorithms require a model that describes images, and operations onto these models like the region merging.

Many works have studied models representing partitions of an image. Topological data structures describe images as a set of elements and their adjacency relations. The most famous example is the Region Adjacency Graph (RAG) [START_REF] Rosenfeld | Adjacency in digital pictures[END_REF] which represents each region by a vertex, and where neighboring regions are connected by an edge. But the RAG suffers from several drawbacks as it does not represent multiple adjacency or makes no differences between inclusion and adjacency relations. To solve these issues, the RAG model has been extended, for instance in dual-graph structure to represent 2D images [START_REF] Kropatsch | Finding the structure of connected components using dual irregular pyramids[END_REF] or in topological maps [START_REF] Bertrand | Topological map: Minimal encoding of 3d segmented images[END_REF][START_REF] Braquelaire | Representation of segmented images with discrete geometric maps[END_REF][START_REF] Damiand | Topological model for two-dimensional image representation: definition and optimal extraction algorithm[END_REF][START_REF] Fiorio | A topologically consistent representation for image analysis: the frontiers topological graph[END_REF] used to represent 2D and 3D images.

The aim of our work is to provide image processing operations using 3D topological maps. We need algorithms that maintain and update the information stored, like relations between regions (adjacency, inclusion, etc.). In a previous work [START_REF] Damiand | Split and merge algorithms defined on topological maps for 3d image segmentation[END_REF], a region merging method has been defined on 3D topological maps, but it is limited to the merging of two adjacent regions.

In this paper, we present and compare two algorithms, one local and one global, for region merging. The first one is dedicated to interactive processing of 3D topological maps as it aims to minimize the number of modifications of the map while merging together a set of connected regions. The global algorithm is designed for automated processing for example in segmentation operations. It allows the merging of any number of regions by connected components. Some experiments show that the two proposed algorithms behave as intended. The local region merging is slower when used intensively contrary to the global region merging.

In Sect. 2, we first present topological maps, which are combinatorial maps verifying specific properties used to represent 3D images. Then, Sect. 3 studies the local approach of region merging in topological maps. We explain the algorithm and give its complexity. In Sect. 4, we present the global approach. Section 5 shows the experimentation results, and compares both approaches in different cases. Lastly, we conclude and give some perspectives in Sect. 6.

Recalls on 3D Topological Maps

A 3D topological map is an extension of a combinatorial map used to represent a 3D image partition. Let us recall the notions on combinatorial maps, 3D images, intervoxel elements and topological maps that are used in this work.

A combinatorial map is a mathematical model describing the subdivision of a space, based on planar maps. A combinatorial map encodes all the cells of the subdivision and all the incidence and adjacency relations between the different cells, and so describe the topology of this space.

The single basic elements used in the definition of combinatorial maps are called darts, and adjacency relations are defined onto darts. We call β i the relation between two darts that describes an adjacency between two i-dimensional cells (see Fig. 1 B for one example of combinatorial map and [START_REF] Lienhardt | Topological models for boundary representation: a comparison with n-dimensional generalized maps[END_REF] for more details on maps and comparison with other combinatorial models). Intuitively, with this model, the notion of cells is represented by a set of darts linked by specific β i relations. For example, a face incident to a dart d is represented by the set of darts accessible using any combination of β 1 and β 3 relations. Moreover, given a dart d, which belongs to an i-cell c, we can find the i-cell adjacent to c along the (i -1)-cell which contains d by using β i (d). For example, given a dart d that belongs to a face f and a volume v, the volume adjacent to v along f is the 3-cell containing β 3 (d).

Let us now present some usual notions about image and intervoxels elements. A voxel is a point of discrete space Z Z 3 associated with a value which could be a color or a gray level. A three dimensional image is a finite set of voxels. In this work, combinatorial maps are used to represent voxel sets having the same labeled value and that are 6-connected. The label of a voxel is given by a labeled function l : Z Z 3 → L that gives for each voxel its label (a value in the finite set L). We speak about region for a maximal set of 6-connected voxel having the same label.

R 1 R 2 R 3
0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1
R 1 R 2 R 3 R 0 A B C D
To avoid particular process for the image border voxels, we consider an infinite region R 0 that surrounds the image. If a region R j is completely surrounded by a region R i we say that R j is included in R i .

In the intervoxel framework [START_REF] Khalimsky | Boundaries in digital planes[END_REF], an image is considered as a subdivision of a 3-dimensional space in a set of cells: voxels are the 3-cells, surfels the 2-cells between two 3-cells, linels the 1-cells between two 2-cells and pointels the 0-cells between two 1-cells (see example in Fig. 1 C where 2-cells, 1-cells and 0-cells are drawn).

The topological map is a data structure used to represent the subdivision of an image into regions. It is composed of three parts:

a minimal combinatorial map representing the topology of the image; -an intervoxel matrix used to retrieve geometrical information associated to the combinatorial map. The intervoxel matrix is called the embedding of the combinatorial map; -an inclusion tree of regions. Figure 1 presents an example of topological map. The 3D image, composed of three regions plus the infinite region R 0 (Fig. 1 A), is represented by the topological map which is divided in three parts labeled B, C and D. The minimal combinatorial map extracted from this image is shown in Fig. 1 B. The embedding of the map is represented in Fig. 1 C, and the inclusion tree of regions in Fig. 1 D.

The combinatorial map allows the representation of all the incidence and adjacency relations between cells of an object. In the topological map framework, we use the combinatorial map as a topological representation of the partition of an image in regions. Each face of the topological map is separating two adjacent regions and two adjacent faces do not separate the same two regions. With these rules, we ensure the minimality (in number of cells) of the topological map (see [START_REF] Damiand | Split and merge algorithms defined on topological maps for 3d image segmentation[END_REF][START_REF] Damiand | Définition et étude d'un modèle topologique minimal de représentation d'images 2d et 3d[END_REF] for more details on topological maps).

The intervoxel matrix is the embedding of the combinatorial map. Each cell of the map is associated with intervoxel elements representing geometrical information of the cell. A face, in the combinatorial map, is embedded by a set of surfels separating voxels of the two incident regions. The edges, which are the border of faces, are represented by a set of linels. The vertices, which are the border of edges, are embedded by pointels. Thus the intervoxel matrix allows to retrieve the geometry of the labeled image represented by the combinatorial map.

The inclusion tree of regions represents the inclusion relations. Each region in the topological map is associated to a node in the inclusion tree. The nodes are linked together by the inclusion relation previously defined. To link the tree with the combinatorial map, each dart d of the map knows its belonging region (called region(d)). Each region R knows one of its dart called representative dart (called rep(R)). rep(R) has to belong to the external surface of R and its other incident region R 2 , given by region(β 3 (rep(R))), has to be a smaller region than R considering the sweeping order of the image voxels (i.e. R 2 is found before R when we run through the image with a scan line algorithm).

Local Region Merging

The objective of the local merging approach is to modify the topological map in a local way to reflect the merging of the selected regions. As its a local operation, we do not want to run through all the darts of the topological map. We also aim to locally transform the inclusion tree of regions. Lastly, it is necessary to respect the minimality property of the topological map and update the geometrical embedding of the map.

Algorithm

The region merging algorithm takes in input a topological map M and a set S of connected regions to merge. The algorithm modifies the topological map M such as all the regions of S are merged together in the resulting region.

Algorithm 1 presents the local approach of the region merging operation. An overview of the local process divides it into three main steps:

initialize the main region and mark the internal faces (i.e. faces incident to two regions that will be merged); -update the inclusion tree to take into account the possible modifications of the inclusion relations; -update the combinatorial map and the embedding by removing internal faces and simplifying their incident edges and vertices.

Before starting the merging process itself, we choose in set S a region, called main region (line 1 of Algo. 1). Instead of creating a new region, we use the main region as the resulting region of the merging algorithm. Thus, we choose the main 2) for the resulting region. The next step is the marking of the internal faces between the regions of set S. We run through the darts belonging to regions of S in the topological map (line 2). Each dart d such as region(d) ∈ S and region(β 3 (d)) ∈ S is incident to an internal face. We mark all the darts incident to the face incident to such a dart. In order to check if a dart belongs to a region of set S in a constant time, we use a second mark applied on regions that belong to S.

To update the inclusion tree (line 4) we remove the selected regions of the tree and then we run through all the darts of the selected regions to find newly internal surfaces. For each one, we build a new connected component of regions in the inclusion tree.

The next step concerns the two other structures of the topological map: the combinatorial map and its embedding, the intervoxel matrix. We want to remove internal faces from the combinatorial map (line 5). This is achieved by using the face-removal (defined in [START_REF] Damiand | Removal and contraction for n-dimensional generalized maps[END_REF] like any other cell-removal operations used in this work) on each internal face previously marked.

The last step of this operation is the map simplification (line 6). Indeed, the removal of the internal faces modifies the degree of the incident edges. Degree two edges are not minimal according to the constraints of the topological map: they are removed with the edge-removal operation. If a degree-two edge is removed, the degree of the two incident vertices changes, so they are simplified if needed.

When a cell is removed in the combinatorial map by using cell-removal operation, the intervoxel matrix is also modified in order to remove the corresponding embedding. For instance, when a face (2-cell) is removed in the combinatorial map, all the corresponding surfels are removed in the embedding.

Complexity

Now we are going to study the time complexity of the local region merging algorithm presented in Algo. 1. The selection of the main region is achieved in O(|S|) where |S| is the number of regions to merge. Checking the validity of the representative dart is a constant time operation.

The loop marking internal faces process each dart belonging to the regions of S exactly once. Testing if region(β 3 (d)) ∈ S is a constant time operation, so this step has time complexity O(D selected ) where D selected is the number of darts belonging to the regions of S.

Updating the inclusion tree is the most expensive operation regarding the number of covered darts. To find new internal surfaces, we run through all the darts belonging to selected regions. This step has complexity O(D selected ). If no inclusion appears, D included , the number of darts belonging to newly included regions, is equal to zero. Otherwise all the included darts are covered. Thus this operation has time complexity O(D selected + D included ) and in the worst case, when all regions are newly included in the resulting region, we cover the darts of the whole topological map.

The cell-removal operation in the combinatorial map has linear time complexity in number of darts incident to the corresponding cell. We only remove cells belonging to the surface of the merged regions. Thus, the complexity is linear in number of darts of the merged region: D selected . Updating the embedding is linear in the number of intervoxel elements representing the cells. The whole process may be upper-bounded by S removed , the number of surfels removed during the face-removal operation (because this number is always greater than the number of linels and the number of pointels). This stage has time complexity O(D selected + S removed ).

Algorithm 1 has time complexity O(D selected + D included + S removed ). It can be upper bounded by the number of darts and the number of surfels of the whole topological map.

Global Region Merging

As a first approach of the region merging in the topological map, the local merge does not provide an effective way to deal with multiple sets of connected regions to merge. To overcome this drawback, we look for another approach allowing several merges at the same time.

The principle of the global merging algorithm is to separate the modifications of the topological map from the merging of regions. The aim is at first to handle the regions at an high level, and then to translate the high level merging into an effective one by removing cells from the topological map.

The first part of the algorithm concerns the symbolic merging of regions. To handle the high level merging and the representation of region sets, we use a disjoint-set forest [START_REF] Cormen | Introduction to Algorithms[END_REF] of regions. Disjoint-sets are used to represent multiple sets. The main operations are the retrieval of the belonging set of an element (find), and the merging of two sets (union). We use union-find trees to represent disjoint-sets. In [START_REF] Tarjan | Efficiency of a good but not linear set union algorithm[END_REF], R. Tarjan shows that the union and find operations can be considered as constant time operations in practical cases.

During the symbolic merging, all regions in a same connected component are merged together in the same union-find tree. Then the merge operation, defined on disjoint-set forest, is used and some internal features of regions like number Rebuild the inclusion tree of regions; 5 of voxels, or mean color are propagated. The root of each tree in the disjoint-set forest will be the resulting region for the merge of the underlying connected set of regions. When all the sets have been processed, each root of the disjoint-set forest is a remaining region of the merging process.

The second part of the algorithm removes all the internal faces. Then, we simplify the topological map and rebuild the inclusion tree.

Algorithm

The global approach of the region merging algorithm takes in input a topological map M and an oracle function that tells how regions have to be merged. The algorithm modifies the topological map M such that all the connected regions that have to be merged according to the oracle are actually merged.

Algorithm 2 presents the global approach of the region merging operation, which is divided into three main steps:

compute the disjoint-set forest of regions: this is the symbolic merging; -remove the internal faces and simplify the incident cells; -build the new inclusion tree.

The first step of the algorithm (line 1 of Algo. 2) concerns the symbolic merging. If the oracle merges r and its neighboring region r 2 in a same connected component, we merge the union-find trees containing the two regions. At the end of the symbolic merging, since only neighboring regions have been merged, each union-find tree represents a connected component of regions. The oracle function could be for example a labelling function which merges regions having the same label.

The next step (line 3) concerns the removal of the internal faces in the topological map. This is the same process as the one used in the internal face removal for the local approach, but without using a mark on internal faces. Indeed, a dart d belongs to an internal face if f ind(region(d)) = f ind(region(β 3 (d))) (i.e. both adjacent regions are in the same disjoint-set). We run through all the darts of the topological map, and for each dart d validating the previous assertion, we use the face-removal operation.

Then, we use the map simplification, presented in the local approach, to obtain the minimal combinatorial map and its corresponding embedding (line 4).

The last step of this approach concerns the building of the new inclusion tree of regions (line 5). This operation is processed with the same algorithm as the one used during the extraction of the topological map (see [START_REF] Damiand | Définition et étude d'un modèle topologique minimal de représentation d'images 2d et 3d[END_REF] for more details). Contrary to the local approach, the inclusion tree is completely destroyed and then rebuilt ; we do not modify the previous tree. This is justified because the global approach is generally used to merge many regions, and in such a case it is more expensive to update the tree than to rebuild it (see experiments in Sect. 5).

Complexity

The first step of the global region merging (presented in Algo. 2) is the symbolic merging. The test line 2 of Algo. 2, which has constant time complexity using a mark on regions, is performed on all the darts of the topological map. The merging of the disjoint-sets containing the two implied regions is considered as a constant time operation. This step has time complexity O(|D|) where |D| is the number of darts of the topological map.

The complexity of the internal face removal is O(|D|) where |D| is the total number of darts in the map. The map simplification algorithm and the updating of the embedding are the same as the ones used in the local approach. So this whole step, covering the face removals and the simplification of the map, has complexity O(|D| + S removed ) where S removed is the number of removed surfels in the embedding. The complexity of the inclusion tree building is in O(|D|) which give us the time complexity of Algo. 2 in O(|D| + S removed ). We can notice that the complexity of the two approaches is the same in the worst case, but they have different processing times depending on the number of merged regions.

Experiments and Analysis

In this section, we are interested in comparing the two approaches of the region merging in topological maps. We study the processing time of the two methods. All the following experiments use as input a same topological map representing 32 3 regions in a 32 3 voxels image (which means each voxel belongs to a different region). In this topological map, we select several regions to merge, in order to study some specific configurations. The goal of the experiments is to compare the behaviour of both algorithms and thus showing the more intresting approach in different cases. For this reason, we use small artificial images to facilitate the comparison without depending on the content of the image.

Merging of a Connected Component of Regions

We have compared the two approaches when the selected regions form only one connected component. We have introduced two protocols of experimentation in this case. For the first one, we have merged an increasing number of regions using both approaches. Figure 2 A shows the total processing time of both methods in function of the number of darts belonging to merged regions. This shows that the processing time of the local approach increases linearly with the number of darts.

The processing time of the global algorithm tends to decrease as the number of merged regions increases. Indeed, during the merging operation, most of the darts are removed, and the simplification and the building of the inclusion tree are cheaper when the number of darts is smaller. To conclude this experiment, we can observe that the local approach is faster than the global one when the number of merged regions remains small. On the contrary, the global approach is faster when the number of regions is bigger. Table 1 presents the processing times of the three different steps of both algorithms in this experiment. The chosen steps are the same as the ones used in the overview of both algorithms. We can observe, in the local approach results, that each step takes an increasing time as the number of merged regions increases. The global approach behave differently. The symbolic merging growth slowly, but the face-removal, the simplification, and the building of the inclusion tree take less time as the number of merged regions increases. These values show the differences between the two approaches and explain the behavior of their processing times.

The second experiment, presented in Fig. 2 B, compares the processing time of the two methods when the number of included regions increases. We have merged a constant number of regions (5768), but these regions are merged in order to include a specified number of regions. This example shows the merging of a small number of regions regarding to the topological map size. This explain why the global approach is slower than the local one in this experiment. The main point we can observe is that the processing time of the global approach remains constant whereas in the local approach where it slowly increases. This is due to the updating of the inclusion tree of regions which depends on the number of included darts. The global approach does not depend on the configuration of these included regions so its processing time is more reliable in general cases.

Merging of Several Connected Components of Regions

We have also compared the two approaches when the selected regions form several connected components. We have studied the impact of the number of sets on the merging process, as well as the impact of their size. We have used the same kind of topological map representing 32 3 regions. In this image, we have selected several regions that belongs to different connected components. To merge such sets of connected regions with the local approach, we have used the local algorithm on each set, and we have computed the total processing time. The same result is obtained by giving to the global method the union of all the sets. Firstly, we have merged an increasing number of sets of connected regions. Each set contains two regions. Figure 3 A presents the processing time of the two approaches in function of the number of merged sets. It shows that the local approach processing time increases with the number of merged set whereas the global method one decreases. The local approach behaves as expected since it is the addition of the processing time of a single merge in the topological map. The global approach behavior is explained, as previously, by the fact that the number of remaining darts decreases and thus the map simplification cost decreases. Figure 3 A shows that the global approach becomes more efficient than the local one as the number of sets growth.

Secondly, we are looking into the influence of the set size on both approaches. In the experiment, we have always merged 256 sets of regions, but we increase the size of each set from 2 to 64 regions. As in the previous experiment, the global merging processing time decreases as the number of merged regions increases. The processing time of the local approach is more expensive when the size of the sets increases. Figure 3 B shows that the global merging operation becomes more efficient than the local one when the size of the connected components increases.

In all the experiments, we have a same conclusion. The local approach is better when the number of selected regions is small and when the number of connected components remains low. This is mainly the case when the region merging is used in an interactive way or when we merge regions during a local process. In the case of a more global process, we prefer to use the global approach of the region merging as it will be generally more efficient and also have a more predictable processing time in function of the image size.

Conclusion

In this paper, we have presented two different approaches of the region merging operation in the 3D topological map. This work shows that the topological map could be efficiently modified in order to represent the evolution of the representation of 3D images during an image processing operation.

We have detailed the local approach of the region merging in 3D topological maps. It processes by applying local modifications to the three components of the topological map: the combinatorial map, the intervoxel matrix and the inclusion tree. The local method allows an efficient processing when the number of merged regions remains small compared to the map size. The configuration of the selected regions also influences the processing time of the region merging when there is included regions. Another drawback of this approach is that the selected regions to merge have to be in only one connected component. If it is not the case, the local region merging have to be applied to each connected component.

To overcome issues of the local approach, we have proposed a second method of region merging. It aims to merge by connected component any number of regions by processing the whole map at once. As shown in the experiments, the processing time does not change even if there are included regions or if there are several connected components. On the contrary, the processing time of the global approach tends to decrease when the number of merged regions increases. This method gives a better way to merge regions in an automated way.

Thus, we have two region merging algorithms on 3D topological maps that allow users to process regions interactively or automatically. These are the first needed tools for image processing with the 3D topological maps. The next step of our work is to use these operations in a real bottom-up segmentation process. Our idea is to change the symbolic merging step of the global approach to merge regions according to a criterion, and then applying the last steps of the algorithm to produce the segmented image. We also want to study the opposite operation of the merge called region splitting. The aim of this operation is to divide a region into several smaller regions given a criterion, which may be a cut surface or an homogeneity measure. A splitting operation will gives the ability to implement a split-and-merge segmentation in the 3D topological map.

Fig. 1 .

 1 Fig. 1. The different parts of the topological map used to represent an image. (A) 3D image. (B) Minimal combinatorial map. (C) Intervoxel matrix (embedding). (D) Inclusion tree of regions.

Algorithm 1 : 3 Update the inclusion tree; 4

 134 Local approach of the region merging operationData: Topological map M ; Connected set of regions S Result: Merge together all the regions of S in M.Choose the main region among the region of S; 1 foreach dart d belonging to regions of S in the map do 2 if region(β3(d)) ∈ S then Mark all the darts belonging to the face incident to d; Remove all internal faces (previously marked); 5 Simplify the cells incident to the removed faces; 6 region so its representative dart respects the definition of the representative dart (see Sect.

Algorithm 2 : 1 if 3

 213 Global approach of the region merging operationData: Topological map M ; Oracle function Result: Merge all the regions by connected components according to Oracle in M. foreach dart d of M do Oracle(region(d),region(β3(d))) then 2 Merge the union-find trees of region(d) and region(β3(d)); Remove all internal faces; Simplify the cells incident to the removed faces;4

Fig. 2 .

 2 Fig. 2. Processing time comparison of the two region merging approaches (processing time given in seconds). (A) By the number of darts belonging to merged regions. (B) By the number of darts belonging to newly included region.

Fig. 3 .

 3 Fig. 3. Processing time comparison of the two region merging approaches (time given in seconds). (A) By the number of sets (size 2) merged. (B) By the size of the 256 sets merged.

Table 1 .

 1 Processing time of different steps of both approaches in function of the number of darts belonging to merged regions.

	Merged darts	24576 196608 393216 589824 786432
	Local approach
	Initialization	0.008 0.052 0.100 0.148 0.196
	Face-removal & simplification 0.028 0.156 0.208 0.316 0.388
	Inclusion tree updating	0.012 0.120 0.304 0.480 0.616
	Total	0.048 0.328 0.612 0.944 1.200
	Global approach
	Initialization	0.060 0.064 0.064 0.072 0.072
	Face-removal & simplification 0.520 0.476 0.424 0.380 0.336
	Inclusion tree building	0.288 0.228 0.152 0.084 0.008
	Total	0.868 0.768 0.640 0.536 0.416

Partially supported by the ANR program ANR-06-MDCA-008-05/FOGRIMMI.


Beatriz Rivas-Murias

Martín Testa-Anta

Pau Torruella

Sònia Estradé

Francesca Peiró

Benito Rodríguez-González

Miguel Comesaña-Hermo

Verónica Salgueiriño

Structural and Magnetic Implications of Transition Metal Migration within Octahedral Core-Shell Nanocrystals

Octahedron-shaped cobalt oxide nanocrystals undergo a structural evolution once coated with thin shells of manganese or cobalt ferrite, by means of an asymmetric solid-solid diffusion occurring at the interface established between the oxides. The resultant mixed ferrites in the final nanostructures stem from the phase progression associated to a non-equilibrium kinetic product that evolves to reach the thermodynamic equilibrium. In this process, the initially strained crystalline lattice closer to the interface influences the progressive redistribution of Co 2+ cations diffusing out of the initial cobalt oxide core, dictating the final magnetic properties. When starting with a non-stoichiometric manganese ferrite shell, the preferential occupation of tetrahedral sites by Mn 2+ cations forces the Co 2+ to occupy octahedral sites, offering a Mn-and Co-doped magnetite shell onto the CoO core.

However, when starting with a cobalt ferrite shell, an extra doping of Co 2+ cations in the strained layers close to the interface forces this ferrite to transition from the inverse to the normal spinel structure, leading to core-shell nanocrystals of CoO and Co-rich cobalt ferrite with an enhanced magnetic moment.

Introduction

The preparation of magnetic nanostructures with ever increasing compositional complexities and exotic morphologies has led to functional materials that are currently implemented in different fields. [START_REF] López-Ortega | Applications of Exchange Coupled Bi-Magnetic Hard/Soft and Soft/Hard Magnetic Core/Shell Nanoparticles[END_REF] More specifically, those of transition metal oxides, with a crystalline ionic configuration of metal cations with unfilled d-electron orbitals, have become key in fields related to energy and magnetism. This stems mainly from the ease of removing and adding electrons to these orbitals that permits different oxidation states, and that is responsible for the vast number of studies in which these materials are implemented in processes related to catalysis, 2,3 chemical looping for batteries, [START_REF] Jang | Direct Synthesis of Self-Assembled Ferrite/Carbon Hybrid Nanosheets for High Performance Lithium-Ion Battery Anodes[END_REF][START_REF] Muhich | Efficient Generation of H2 by Splitting Water with an Isothermal Redox Cycle[END_REF][START_REF] Bayliss | Probing Mg Migration in Spinel Oxides[END_REF] and spintronics. [START_REF] Ramos | Influence of Oxidation on the Spin-Filtering Properties of CoFe2O4 and the Resultant Spin Polarization[END_REF][START_REF] Lukashev | Interface States in CoFe2O4 Spin-Filter Tunnel Junctions[END_REF] The reduced dimensionality of magnetic nanostructures can result in significant changes in the magnetic behavior, [START_REF] Carta | A Structural and Magnetic Investigation of the Inversion Degree in Ferrite Nanocrystals MFe2O4 (M = Mn, Co, Ni)[END_REF][START_REF] Peddis | Cationic Distribution and Spin Canting in CoFe2O4 Nanoparticles[END_REF] not only owing to the highly correlated d electrons mentioned, [START_REF] Dionne | Magnetic Oxides[END_REF] but also due to confinement effects, [START_REF] Huijben | Critical Thickness and Orbital Ordering in Ultrathin La0.7Sr0.3MnO3 Films[END_REF][START_REF] Moyer | Enhanced Magnetic Moment in Ultrathin Fe-Doped CoFe 2O 4 Films[END_REF][START_REF] Rigato | The Magnetization of Epitaxial Nanometric CoFe2O4 (001) Layers[END_REF] which can be taken into consideration at interfaces created when combining these oxides with low dimensionalities. Along these lines, an interface embodies an interesting source of relevant structural effects (auxetic to non-auxetic crossover) [START_REF] Ferreiro-Vila | Apparent Auxetic to Non-Auxetic Crossover Driven by Co 2+ Redistribution in CoFe2O4 Thin Films[END_REF] or magnetic interactions (spin canting, exchange bias or charge transfer) [START_REF] Testa-Anta | Spin Frustration Drives Exchange Bias Sign Crossover in CoFe2O4-Cr2O3 Nanocomposites[END_REF] that can lead to synergies and eventually to a completely new set of electronic/magnetic features. [START_REF] Zubko | Interface Physics in Complex Oxide Heterostructures[END_REF] Moreover, interfacial phenomena can offer a way to reach phases that are thermodynamically stable but cannot be synthesized experimentally, by means of a solid state chemistry path through multiple non-equilibrium intermediates. [START_REF] Bianchini | The Interplay between Thermodynamics and Kinetics in the Solid-State Synthesis of Layered Oxides[END_REF] Indeed, the use of interfaces as the starting point in preparative solid-state chemistry takes into account an evolution of the metastable compounds located in their surroundings towards a situation of local minima in the thermodynamic landscape. Consequently, an original mechanistic pathway can be explored for the synthesis of new solids and a deeper understanding of their ultimate functions, 19,20- 22 providing therefore an unprecedented degree of control when referred to the magnetic properties of transition metal oxides at the nanoscale.

Particularly, the magnetic nanostructures of 3d-transition metal oxides, porous and/or hollow, are gaining relevance due to their low densities, large surface areas and reduced length for both mass and charge transport properties. Such characteristics are currently implemented to maximize performances in fields related to Li-ion batteries, theranostics or catalysis. [START_REF] López-Ortega | Galvanic Replacement onto Complex Metal-Oxide Nanoparticles: Impact of Water or Other Oxidizers in the Formation of Either Fully Dense Onion-like or Multicomponent Hollow MnOx/FeOx Structures[END_REF][START_REF] Yu | Complex Hollow Nanostructures: Synthesis and Energy-Related Applications[END_REF][START_REF] Wei | Biodegradable and Renal-Clearable Hollow Porous Iron Oxide Nanoboxes for in Vivo Imaging[END_REF][START_REF] Boebinger | Spontaneous and Reversible Hollowing of Alloy Anode Nanocrystals for Stable Battery Cycling[END_REF] In opposition to the use of sacrificial templates for the formation of hollow structures, [START_REF] Vaz | Enhancing the Exploitation of Functional Nanomaterials through Spatial Confinement: The Case of Inorganic Submicrometer Capsules[END_REF] there are different diffusiondriven processes, for example taking advantage of the Kirkendall effect [START_REF] Kirkendall | Diffusion of Zinc in Alpha Brass[END_REF][START_REF] Smigelkas | Zinc Diffusion in Alpha Brass[END_REF][START_REF] Yin | Formation of Hollow Nanocrystals Through the Nanoscale Kirkendall Effect[END_REF][START_REF] Wang | Hollow Nanocrystals through the Nanoscale Kirkendall Effect[END_REF] or galvanic transformations [START_REF] López-Ortega | Galvanic Replacement onto Complex Metal-Oxide Nanoparticles: Impact of Water or Other Oxidizers in the Formation of Either Fully Dense Onion-like or Multicomponent Hollow MnOx/FeOx Structures[END_REF][START_REF] Oh | Galvanic Replacement Reactions in Metal Oxide Nanocrystals[END_REF] that have proven to be particularly useful. In fact, solid-solid diffusion processes imply a privileged strategy because the atoms and vacancies that form these crystalline solids are not static but rather mobile entities that can experience dynamic processes. [START_REF] Gusak | Diffusion-Controlled Solid State Reactions: In Alloys, Thin Films and Nanosystems[END_REF] Such phenomena endow the hybrid systems in consideration with rich chemical characteristics that have an important impact in the magnetic properties derived. Moreover, they permit the formation of equilibrium stoichiometries, holding the system under appropriate thermodynamic conditions (temperature and pressure) for a sufficiently long time. [START_REF] Bianchini | The Interplay between Thermodynamics and Kinetics in the Solid-State Synthesis of Layered Oxides[END_REF] In the present work, we detail the structural and magnetic characterization of transition metal oxide nanostructures that result from the coating of cobalt oxide octahedron-shaped nanocrystals with a thin shell of manganese or cobalt ferrite by means of a thermal decomposition process. This study reveals a structural evolution stemming from a solid-solid diffusion process starting at the core-shell interface, which directs the magnetic properties of the final hybrid material. The mechanism proposed is supported by a comprehensive and multi-instrumental study considering Raman spectroscopy and advanced electron microscopy techniques, jointly with magnetic measurements, which permits an indepth characterization of the octahedron-shaped final nanostructures and allows us to correlate the changes of the total magnetic moment with the reconfiguration of the magnetic cations in the spinel structure. To explain the mixing and hollowing process observed, we have taken into account the cation vacancies and interstitials as the most important point defects, while considering that oxygen diffusion is always a much slower process to drive and influence the solid-solid diffusion itself. [START_REF] Lu | Point Defects and Cation Tracer Diffusion in (Co, Fe, Mn)3-ΔO4 Spinels: I. Mixed Spinels (CoxFe2yMny)3-O4[END_REF] Experimental Synthesis of octahedron-shaped cobalt oxide nanoparticles. Cobalt(II) acetate tetrahydrate (1 mmol), trioctylphosphine oxide (TOPO, 0.2 mmol), oleic acid (8 mmol) and oleylamine (14.3 mmol) were added into a three-necked round-bottomed flask containing 25 mL of trioctylamine (TOA), in open air conditions. The temperature was increased up to 140 °C (6 °C/min ramp rate) and then kept at this temperature with magnetic stirring for 45 min. After this dwell period, the solution was again heated to reflux up to 300 °C (3 °C/min ramp rate). Upon reaching the reflux, the solution was kept at this temperature for 5 h and subsequently cooled down to room temperature. Finally, the nanoparticles were washed by centrifugation and redispersed in ethanol.

Synthesis of CoO@MnFe2O4 (CoO@MFO) and CoO@CoFe2O4 (CoO@CFO) core-shell octahedra. For this synthesis, the previously synthesized octahedron-shaped cobalt oxide nanoparticles were used as seeds for the thermal decomposition of iron (II) and manganese (II) or iron (II) and cobalt (II) acetate precursors by means of a hot-injection procedure. Thus, Thus, 20 mg of the pre-synthesized cobalt oxide nanoparticles (0.267 mmol) of the pre-synthesized cobalt oxide nanoparticles (as solid material), 1.6 mmol of oleic acid and 25 mL of TOA were placed in a threenecked round-bottomed flask and heated up to 210 °C under stirring. Once this temperature was reached, a 500 μL solution including 19.16 μmol of iron (II) acetate and 9.58 μmol of manganese (II) or cobalt(II) acetate dissolved in ethanol was injected dropwise, trying to avoid an important decrease in temperature. After that, the solution was heated up to reflux (300 °C, 3 °C/min ramp rate) and kept at this temperature for 3 h. Upon heating, the solution changed color from brown to black. Once cooled to room temperature, ethanol (25 mL) was added into both solutions to precipitate nanoparticles, helped by centrifugation. The nanoparticles were thus separated from the supernatant containing the solvent, ligands and by-products. This procedure was repeated at least three times.

Characterization. Samples for transmission electron microscopy (TEM) analysis were prepared by drop casting a diluted suspension of the nanoparticles onto an ultrathin carbon coated copper grid. TEM measurements were performed on a JEOL JEM1010 instrument operating at an acceleration voltage of 100 kV. High resolution transmission electron microscopy (HRTEM), high-angle annular dark-field scanning transmission electron microscopy (HAADF STEM) and energy dispersive X-ray elemental analyses (EDS) were carried out on a field emission JEOL JEM2010F TEM working at 200 kV. STEM-HAADF imaging, electron energy loss spectroscopy (EELS), HAADF-tomography and EELS-tomography were performed in a JEOL ARM200F equipped with a Quantum GIF spectrometer. Inspect3D software was used for the tomography reconstruction. X-ray diffraction patterns were collected using a Panalytical X'Pert Pro diffractometer (Cu Kα radiation and Bragg-Brentano -2 geometry) in the 10-80° 2θ range. Raman spectra were collected with a Renishaw in Via Reflex Confocal Raman Microscope. These experiments were registered in backscattering geometry at room temperature using a 785 nm laser excitation wavelength. The laser beam was focused on the sample by a 20x objective, with a numerical aperture (NA) value of 0.40. The laser powers used were below 1 mW to avoid overheating on the samples. Magnetic measurements were performed using the VSM option in a Physical Property Measurement System (PPMS) from Quantum Design. Hysteresis loops were measured at 10 and 325 K up to an external field of 7 T. FC hysteresis loops were performed at 10 K, applying a magnetic field of 5 T in the cooling process from 340 K.

The magnetization dependence with temperature in zero-field-cooled (ZFC) and field-cooled (FC) conditions was performed at 10 mT in the 5-330 K range. All magnetic measurements were performed using dried powdered samples.

Results and Discussion

Figure 1 includes low and high magnification TEM images with general overviews of the three samples considered in this study, starting with octahedron-shaped cobalt oxide nanoparticles (hereafter labelled as CoO, figures 1a and 1b), subsequently coated with a thin layer of manganese ferrite (hereafter labelled as CoO@MFO, figures 1d and 1e) or cobalt ferrite (hereafter labelled as CoO@CFO, figures 1g and 1h), obtained in independent experiments. In the three cases we can appreciate the octahedral shape, perfectly defined in the initial CoO nanocrystals with flat surface facets that subsequently evolve to more irregular morphologies for the CoO@MFO and CoO@CFO samples, respectively. Likewise, the nanocrystals display cracks in their interior which become gradually more relevant for the CoO@MFO and CoO@CFO systems, respectively. Additionally, the histograms plotting the particle size distribution (i.e., the edge length distribution) and their fit to a log-normal function for the three samples are shown, with average edge length values l = 50.4 */ 1.6 nm (for the CoO sample, figure 1c), l = 41.9 */ 1.5 nm (for the CoO@MFO sample, figure 1f) and l = 37.3 */ 1.4 nm (for the CoO@CFO sample, figure 1i), attending to the fact that it is more accurate to reflect the size distribution in terms of a log-normal function, with the geometric mean multiplied or divided by the square of the standard deviation (𝑙 = x ̅ * * /(s * ) 2 nm, containing 95.5 % of the nanoparticles). [START_REF] Limpert | Log-Normal Distributions across the Sciences: Keys and Clues[END_REF] Surprisingly, coating the initial CoO octahedra either with manganese or cobalt ferrite leads to a shrinkage of the average size of the final nanocrystals. To understand the shrinkage of the nanocrystals once coated, we have studied the internal morphology of the CoO@MFO and CoO@CFO samples. Figures 2a and2b include HAADF images (in STEM mode) of these nanostructures, revealing the presence of defects or voids as regions of lower density, despite the fact of maintaining in most of the cases the octahedral shape and the faceted outer surface. These voids, present in the CoO initial sample, become larger in the CoO@MFO and CoO@CFO samples, implying therefore additional solid-state diffusion processes and intermixing. [START_REF] Kohlmann | Looking into the Black Box of Solid-State Synthesis[END_REF][START_REF] Oh | Galvanic Replacement Reactions in Metal Oxide Nanocrystals[END_REF][START_REF] Moon | Chemical Transformations of Nanostructured Materials[END_REF][START_REF] Anderson | Nanoparticle Conversion Chemistry: Kirkendall Effect, Galvanic Exchange, and Anion Exchange[END_REF] Hence, first, in order to obtain an in-depth understanding of the structural features of these nanostructures, Raman spectroscopy was employed. This technique has proven to be particularly useful in those cases in which chemical segregation and/or cation doping lead to the formation of complex interfaces in oxides. [START_REF] Testa-Anta | Raman Spectroscopy to Unravel the Magnetic Properties of Iron Oxide Nanocrystals for Bio-Related Applications[END_REF] According to the literature, while CoO, which is a weak Raman scatterer, shows a broad band (one-phonon LO mode) at 530 cm -1 , [START_REF] Rivas-Murias | Thermodynamic CoO-Co3O4 Crossover Using Raman Spectroscopy in Magnetic Octahedron-Shaped Nanocrystals[END_REF][START_REF] Melendres | A Laser Raman Spectroscopic Study of Anodic Corrosion Films on Silver and Copper[END_REF][START_REF] Ravindra | Laser Induced Structural Phase Transformation of Cobalt Oxides Nanostructures[END_REF] the general AB2O4 spinel lattice exhibits five Raman-active modes (one A1g, one Eg, and three T2g modes). [START_REF] White | Interpretation of the Vibrational Spectra of Spinels[END_REF][START_REF] Verble | Temperature-Dependent Light-Scattering Studies of the Verwey Transition and Electronic Disorder in Magnetite[END_REF] The highest-frequency A1g mode of the spinel structure, attributed to the symmetric stretching of oxygen atoms ("breathing motion") in the tetrahedral AO4 void [START_REF] Cynn | High-Temperature Raman Investigation of Order-Disorder Behavior in the MgAl2O4 Spinel[END_REF][START_REF] D'ippolito | Raman Fingerprint of Chromate, Aluminate and Ferrite Spinels[END_REF][START_REF] Shebanova | Raman Spectroscopic Study of Magnetite(FeFe2O4): A New Assignment for the Vibrational Spectrum[END_REF] along the cubic111 direction, [START_REF] Iliev | Monitoring B-Site Ordering and Strain Relaxation in NiFe2O4 Epitaxial Films by Polarized Raman Spectroscopy[END_REF] is typically the most intense and is usually located at approximately 600-750 cm -1 . The remaining lowfrequency modes also reflect a rather strong influence of the tetrahedral sites, although these units are not completely isolated but surrounded by three octahedral units. This last fact renders Raman spectroscopy a valuable tool to ascertain different cationic rearrangements within the spinel structure.

Figure 2c displays the Raman spectra registered from the CoO@MFO (blue) and the CoO@CFO (red) samples, besides the one from the CoO sample as reference (in black), which shows its characteristic band at 533 cm -1 (shadowed in grey). The spectrum of the CoO sample displays an additional band at 690 cm -1 , which can be associated to the native oxidation of CoO to Co3O4 at the surface of the octahedra (the main T2g, Eg and A1g bands of the spinel Co3O4 appear shadowed in yellow in the graph). [START_REF] Rivas-Murias | Thermodynamic CoO-Co3O4 Crossover Using Raman Spectroscopy in Magnetic Octahedron-Shaped Nanocrystals[END_REF][START_REF] Fontaíña-Troitiño | Room-Temperature Ferromagnetism in Antiferromagnetic Cobalt Oxide Nanooctahedra[END_REF][START_REF] Li | Electrostatic Doping as a Source for Robust Ferromagnetism at the Interface between Antiferromagnetic Cobalt Oxides[END_REF] Considering first the CoO@MFO nanocrystals, besides the distinctive feature from CoO, three additional bands centered at 192, 484 and 691 cm -1 (shadowed in yellow) can be observed, indicating the presence of Co3O4 within the nanocrystals. The presence of a manganese-based ferrite can be also evidenced by the two small shoulders located at 630 and 658 cm -1 (indicated by blue dotted lines in figure 2c). Whereas the A1g mode from the MnFe2O4 phase is usually located in the 600-625 cm -1 region, the characteristic A1g mode from magnetite is known to appear at ~660-670 cm -1 . The observation of the mode at 630 cm -1 proves therefore the insertion of Mn 2+ cations into the tetrahedral sites of a spinel ferrite structure. Likewise, the fact that we have registered a second mode at 658 cm -1 indicates the presence of manganese-doped magnetite, though not homogeneous throughout the sample, as confirmed by scanning different spots (see figure S1 in the Supporting Information, SI). Regarding the CoO@CFO nanocrystals, the presence of CoO can be analogously evidenced from the main feature at 540 cm -1 , whereas the bands located at 470 and 650 cm -1 can be presumably ascribed to the T2g(2) and A1g phonon modes of the CoFe2O4 spinel structure, respectively (indicated by red dotted lines). Due to the presence of two different cations at the octahedral sites of its inverse spinel structure, CoFe2O4 typically displays a splitting of the A1g mode, leading to one intense mode accompanied by a small shoulder at lower frequencies. Interestingly, the more intense A1g mode (A1g (2)) appears shifted if compared to the values reported in the literature for the stoichiometric inverse cobalt ferrite. Given the fact that this vibrational mode is highly sensitive to cation redistribution, [START_REF] Chandramohan | Cation Distribution and Particle Size Effect on Raman Spectrum of CoFe2O4[END_REF][START_REF] Bahlawane | Tailoring the Properties and the Reactivity of the Spinel Cobalt Oxide[END_REF] this redshift observed points to a scenario in which a cationic rearrangement has taken place, leading to a cobalt-rich ferrite shell. Even though the presence of tensile strain within the attained nanostructures may also contribute to the A1g redshift, the fact that the remaining modes remain unshifted evidences an intermixing process as the main driving force. This experimental Raman analysis, jointly with the observations of the HAADF-STEM images, suggest simultaneous and progressive inwards and outwards cation diffusion processes that cause a shrink of the octahedra average size while rendering the nanostructures porous/hollow. Figure S2 (in the SI) includes the powder diffractograms of CoO@MFO and CoO@CFO, showing that the presence of the rock-salt CoO crystalline structure screens the spinel structure registered by Raman spectroscopy in both samples. Diffracted peaks associated to the spinel structure with very low intensities (indicated by black arrows) and a peak associated to metallic iron (bcc structure) were also registered. Since starting with nanocrystals with voids and cracks inside, this can also imply that each nanostructure is polycrystalline, as suggested from the XRD patterns of the samples. Indeed, when applying the Scherrer equation to the (111) diffracted peak, an average  23 nm crystallite size is obtained. Having the intermixing process in mind, a more detailed HRTEM analysis combined with EELS mapping and tomography reconstruction was performed for the CoO@MFO and CoO@CFO nanocrystals. Figure 3 shows the EELS analysis of a representative nanocrystal from the CoO@MFO sample (top blue panel) and from the CoO@CFO sample (bottom red panel). For each image, the Co (780 eV), Fe (710 eV), Mn (640 eV) and O (530 eV) peaks have been mapped. In both cases, the Fe elemental maps clearly reveal the presence of an iron rich layer of 3 nm on the surface of the nanocrystal (figures 3b and 3f of the blue panel for the CoO@MFO sample and figures 3b and 3e of the red panel for the CoO@CFO sample), which we associate to the spinel ferrite shells formed and registered by Raman spectroscopy. For the CoO@MFO sample, the manganese present could only be detected after the use of a sophisticated statistical analysis (PCA, Principal Components Analysis), since its relative composition is below 5%. The composition analysis for the outer shell of this sample indicates however the formation of a material richer in cobalt, with an approximate average Mn0.4Co1.8Fe0.8O4 spinel stoichiometry. Note that although the manganese signal image looks relatively clear, the intensity is an order of magnitude lower than that of iron or cobalt. For the CoO@CFO sample, the EELS analysis of the outer shell reveals an average Co1.9Fe1.1O4 composition, assuming again the spinel crystalline structure. These stoichiometric ratios for the spinel shell in both samples are averaged over the whole shells, but the fact of observing clear iron-rich outer areas (see figures 3b and 3f in the blue panel and figures 3b and 3e in the red panel) points more to a negative gradient of cobalt as moving in the outward direction. Furthermore, a 3D tomographic reconstruction of the nanocrystals obtained from the EELS signal collected was carried out, in order to obtain a more graphical picture of the internal morphology.

Figures 4a and4b depict the representative HAADF tomography images of two different nanocrystals from the CoO@MFO sample. The reconstruction using the cobalt signal is shown in figures 4c and 4d, on which we can appreciate the edges of the octahedron shape highlighted with black dotted lines (figure 4c) or an internal channel up to the surface of the nanocrystal (figure 4d). Since the iron and manganese signals were too low to produce a reliable 3D reconstruction, we have taken advantage of a 3D reconstruction based on the oxygen signal (figures 4e and 4f). These 3D reconstructions depict the nanocrystals with a larger volume than in figures 4c and 4d, which means that there is a uniform outer layer of an oxide at the surface of the nanocrystal, corroborated also with the previous EELS analyses shown in figure 3 (blue panel). When plotting both reconstructions together (i.e. considering the cobalt and oxygen signals), as shown in figures 4g and 4h, a clear image of the core-shell structure emerges. Analogously, figure 5 includes the 3D tomographic reconstruction of a nanocrystal from the CoO@CFO sample, which confirms a more advanced deformation of the original octahedral shape. Nevertheless, eight triangular facets can still be observed (highlighted with black dotted lines, figure 5b) together with the presence of voids and channels within the structure. Indeed, one of the surface facets shows an aperture that is attributed to the presence of one of the channels towards the surface (seen in figures 5a, 5d, 5e, 5g and 5j). The bulk of the particle mainly consists of cobalt-based oxide, with the iron seemingly distributed throughout the structure following a complex grid of channels and small voids, supporting the idea that solid-solid diffusion processes took place. Such finding also highlights the presence of a 3 nm outer iron-based layer, as clearly reflected in this 3D reconstruction. Moreover, the presence of metallic iron traces (as evidenced in figure 5l) can be explained by the formation of wüstite as by-product during the process, which evolves at high temperatures to magnetite and metallic iron. oxygen (e and f), or both elements together (g and h) (cobalt in green and oxygen in cyan blue). Given the very low concentration of manganese in the CoO@MFO nanocrystals, an additional evaluation of the elemental distribution was also carried out by EELS point analyses together with further HRTEM studies. Figure 6 includes the summary of the analysis of a nanocrystal from the CoO@MFO sample, with a large cavity inside (see figure 6a) and with a core-shell structure (see figures 6b and 6d). Figure 6c shows the corresponding fast Fourier transform (FFT), on which the indexation indicated in red was done on the basis of the spinel crystalline structure and the indexation in yellow was done on the basis of the CoO rock-salt crystalline structure, both indicating the [211] zone axis. The lack of evident extra spots or double diffraction spots in the FFT image can be explained if we assume a directed epitaxial growth of the mixed spinel onto the rock-salt crystalline structure. In this regard, figure 6d includes a HRTEM image of the edge of this hollow nanoparticle which reflects this expected core-shell structure, clearly showing that there are two different crystalline phases (lattices), with the second one (the shell thickness is about ~ 3.2 nm matching the 3D tomography results) grown driven by the {111} surface facets of the initial cobalt oxide octahedron. This TEM analysis also reveals that the MnxFe3-xO4 grains have grown, guided by the lattice matching along the eight {111} surface planes of the core, but leading to a fragmentation of the shell into multigrains. [START_REF] Oh | Design and Synthesis of Multigrain Nanocrystals via Geometric Misfit Strain[END_REF] In this regard, the fact that every initial cobalt oxide core presents cracks up to the surface also helps promoting the formation of an ordered but grained structure, as shown in figure 6b. Furthermore, figures 6e and 6f show the EELS spectra spatially resolved after background subtraction at the O-K, Mn-L2,3, Fe-L2,3, and Co-L2,3 edges. They were collected on different spots of the nanoscrystal shown in figure 6a, over the large region framed in green in the inset of figure 6e.

Focusing first on the O-K edge (see a more detailed comparison in figure S3 in the SI), the fine structure can distinguish the different cobalt oxide phases present in the sample due to its sensitivity to the chemical environment. The intense and single peak at 535 eV, characteristic of the presence of CoO, [START_REF] Fontaíña-Troitiño | Room-Temperature Ferromagnetism in Antiferromagnetic Cobalt Oxide Nanooctahedra[END_REF][START_REF] Barreca | Controlled Vapor-Phase Synthesis of Cobalt Oxide Nanomaterials with Tuned Composition and Spatial Organization[END_REF] and the L3/L2 ratio for the Co-L edge (also indicative of the cobalt valence state) [START_REF] Barreca | Controlled Vapor-Phase Synthesis of Cobalt Oxide Nanomaterials with Tuned Composition and Spatial Organization[END_REF][START_REF] Zhang | Surface Effects in the Energy Loss near Edge Structure of Different Cobalt Oxides[END_REF][START_REF] Pearson | Measurements of 3d State Occupancy in Transition Metals Using Electron Energy Loss Spectrometry[END_REF][START_REF] Wang | EELS Analysis of Cation Valence States and Oxygen Vacancies in Magnetic Oxides[END_REF] point to the CoO as the main phase. However, the spectra shown in figure 6f, collected with a 1 nm probe size from the red and blue spots shown in the HRTEM image in figure 6a, reflect a different composition at the core and the shell. Hence, the presence of Mn and Fe mainly at the external layer confirms the core-shell structure shown in the HRTEM analysis. To complete the characterization, figure 7 and table I include a summary of the magnetic behavior of the three samples considered. The temperature dependence of the magnetization under ZFC and FC conditions (0H = 10 mT) was measured, showing irreversibility between the ZFC and FC curves up to the highest measured temperature (T = 325 K), which is above the Néel temperature of CoO (TN  291 K for bulk, indicated in the ZFC-FC graphs with a vertical dotted line). Indeed, the CoO sample shows a very subtle peak in this region (indicated with a black arrow in figure 7a, andclearly shown in the derivative of the temperature-dependent magnetization curve included in figure S4, in the SI), related to its antiferromagnetic-paramagnetic transition. This transition can also be observed for the CoO@MFO sample (figure 7d, indicated with a blue arrow and in figure S4 in the SI), but screened for the CoO@CFO sample. At lower temperatures, additional transitions are present in the CoO sample, which can be attributed to the outer Co3O4 shell (TN  40 K) or to a blocking temperature of smaller nanocrystals. Due to the presence of the ferrimagnetic ferrite components, the coated systems display much larger values of magnetization, with a noteworthy difference between the ZFC and FC curves throughout the whole range considered. As in the case of CoO, the features observed at very low temperatures in these two samples are indicative of the presence of Co3O4 and smaller nanocrystallites.

The hysteresis loops from the three samples were registered at 10 and 325 K, showing in the three cases the coexistence of a ferrimagnetic (hysteresis) and an antiferromagnetic contribution (linear response at high fields in the coated samples). The ferrimagnetic contribution is responsible for the increase in the values of maximum magnetization, from roughly 5.7 Am 2 /kg in the non-coated sample to 40 and 80 Am 2 /kg at 10 K in the CoO@MFO and CoO@CFO samples, respectively. The values of saturation magnetization upon subtraction of the linear antiferromagnetic contribution are 35 and 78 Am 2 /kg at 10 K, and 30 and 74 Am 2 /kg at 325 K for the CoO@MFO and CoO@CFO samples, respectively. The field-dependent magnetization of the three samples was also registered at 10 K after field-cooling the samples. Figure S5 in the Supporting Information includes a comparison of the 13.9 16.9 -24.9

CoO@MFO 30

CoO@CFO 74 78 10.1

In order to correlate the magnetization values with the morphology attained, we have calculated the expected maximum magnetization (at 10 K) of an ideal CoO octahedral particle with a 50 nm edge (considering a maximum value of magnetization of 5.7 Am 2 /kg, as registered experimentally), coated with a perfect 3 nm-thick shell of stoichiometric MnFe2O4 or CoFe2O4, (considering saturation magnetizations of 112 and 90 Am 2 /kg, respectively), [START_REF] Cullity | Introduction to Magnetic Materials[END_REF] using tabulated mass densities and neglecting any other contribution to the magnetization. Taking this ideal morphology into account, we obtain values of maximum magnetization of 32 Am 2 /kg and 27 Am 2 /kg for perfect CoO@MFO and CoO@CFO core-shell octahedra. These values are far below the experimental ones obtained at 10 K and applying a 7 T magnetic field. Going one step further, we proceed with the same calculation, but considering the average shrinkage observed experimentally and a theoretical 10% or 15% of voids in the total volume of the 35.9 nm CoO belonging to the CoO@MFO or of the 31.3 nm CoO belonging to the CoO@CFO, respectively. This way we also assume the formation of an average 3 nm thick perfect shell of Mn0.4Fe2.6O4 (MS = 103.6 Am 2 /kg) or CoFe2O4 (MS = 90 Am 2 /kg). With this situation into account, having considered the more important shrinkage in the final size, the larger dimensions of the hollow cavities inside the nanocrystals of the CoO@CFO sample, and iterating the contribution from metallic iron to approach the experimental values (contribution of 3 vol.%), maximum magnetization values of 40 Am 2 /kg for the two CoO@MFO and CoO@CFO samples were obtained.

Such values are close to the experimental data obtained for the CoO@MFO but far below the value measured for the CoO@CFO sample, which led us to think of a different reason besides the presence of metallic iron for the magnetic moment enhancement in this sample, as actually including, according to the XRD intensities, an even lower amount of the metal in this sample.

To shed light on this and summarizing the characterization section, we can state that the core-shell CoO@MFe2O4 (M: Mn or Co) nanoparticles undergo a restructuring-hollowing process by which there is an important shrinkage in the final size and a significant increase in the dimension of the hollow cavities inside the nanocrystals. In this manner, the morphological evolution of the nanoparticles is the result of solid-solid diffusion processes by which inhomogeneous mixed oxides are formed. Accordingly, the nanocrystals in the CoO@MFO sample include CoO, Co3O4 and Mnand Co-doped Fe3O4, with a negative gradient in the cobalt doping as moving in the outward direction and the presence of residual metallic iron. Analogously, the nanocrystals from the CoO@CFO sample include CoO and Co3O4 (residual), Co-doped CoFe2O4 (cobalt-rich layer) and Co-doped Fe3O4 (ironrich layer) with some residual metallic iron, also following the outward direction. The presence of a cobalt-rich spinel is indeed corroborated by the soft behavior observed for this sample at low temperature, with a coercivity value of 0HC 22 mT (see summary of magnetic parameters of the three samples in table I, in the SI), a rather small value if one takes into account the hard magnetic behavior displayed by the stoichiometric inverse CoFe2O4. [START_REF] Sharifi Dehsari | Impact of Stoichiometry and Size on the Magnetic Properties of Cobalt Ferrite Nanoparticles[END_REF] Furthermore, this small value of coercivity confirms as well the residual amount of metallic iron. The origin of such soft behavior can be ascribed to the redistribution of Co 2+ ions among the octahedral/tetrahedral sites and the Fe/Co ratio gradient, which change the effective magnetic anisotropy of the system due to Co 2+ ions in the trigonal-field at the octahedral sites.

To understand this final morphology attained and the disposition of stratified layers of different oxides in the final nanocrystals, with a magnetically dominant spinel structure in view of the magnetic properties they display, different aspects must be taken into account. First, this final distribution is postulated to appear once the growth process has finished but the system has not attained its thermodynamic stability yet. At this point, the surface energy of a material at the nanoscale can alter its free energy, such that another material with the same elements but different stoichiometry and crystalline structure is formed, driven by a thermodynamic crossover in phase stability. [START_REF] Navrotsky | Nanophase Transition Metal Oxides Show Large Thermodynamically Driven Shifts in Oxidation-Reduction Equilibria[END_REF] Such phase reconfiguration is indeed favored given the relatively high temperature of the synthetic process and assisted by effects stemming from the nanometer size. In this regard, a chemomechanical model of void formation has been recently reported, by which the mechanical constraint of an oxide layer during a delithiation process causes nanoparticles to form voids or to allow for particle shrinkage without void formation, depending on the particle size. [START_REF] Boebinger | Spontaneous and Reversible Hollowing of Alloy Anode Nanocrystals for Stable Battery Cycling[END_REF] Analogously, in our case, as Co 2+ cations diffuse out from the core, the CoO phase shrinks in volume, exerting a compressive stress on the outer ferrite layer. In response, the ferrite layer applies increasing tension to the core, with a situation similar to a pressure vessel with negative pressure, in the early stages of the process. [START_REF] Boebinger | Spontaneous and Reversible Hollowing of Alloy Anode Nanocrystals for Stable Battery Cycling[END_REF] Attending to this model, the buckling of the ferrite layer to allow for particle shrinkage is more favored, since relatively large particles require a greater strain energy to create new voids. Moreover, the assynthesized cobalt oxide octahedra are already endowed with voids and cracks (see TEM image in figure 1b), [START_REF] Ramos-Docampo | Thermodynamically Driven Oxidation-Induced Kirkendall Effect in Octahedron-Shaped Cobalt Oxide Nanocrystals[END_REF] which promote simultaneously with the particle shrinkage, the hollowing process. From the point of view of redox processes, galvanic reactions may also take place but to a much lesser extent. Indeed, the ferrite oxidation reactions are cation diffusion-limited under oxygen rich environments, because of the larger concentration of cationic vacancies with respect to anionic ones (oxygen) in the crystalline structure. [START_REF] Lu | Point Defects and Cation Tracer Diffusion in (Co, Fe, Mn)3-ΔO4 Spinels: I. Mixed Spinels (CoxFe2yMny)3-O4[END_REF][START_REF] Lu | Point Defects and Cation Tracer Diffusion in (CoxMn1-x)3-O4 Spinels[END_REF][START_REF] Dieckmann | Defects and Cation Diffusion in Magnetite (II). Berichte der Bunsengesellschaft/Physical[END_REF] In this context, the crystal field theory has been used to explain relative cationic diffusion preferences, taking into account the degree of tetrahedral/octahedral site preference and the consequent degree of inversion. [START_REF] Miller | Distribution of Cations in Spinels[END_REF] For example, activation energies for the vacancy mediated diffusion of cations in different ferrites, considering atomic and spin configurations were calculated for both normal and inverse configurations in the crystalline structure and the magnetic behavior. [START_REF] Muhich | First-Principles Analysis of Cation Diffusion in Mixed Metal Ferrite Spinels[END_REF] According to such calculations, the cationic vacancies do not only occupy preferentially octahedral over tetrahedral sites but also indicate that cation diffusion through ferrites is dominated by octahedral-octahedral hops, [START_REF] Peterson | Correlation and Isotope Effects for Cation Diffusion in Magnetite[END_REF] limiting vacancy exchange between the two sites. This reaction pathway for octahedral-octahedral migration considers activation energies resulting from the occupation and energies of the t2g and eg states of the active cation, by which the final magnetic state and the magnetic properties are governed. Along these lines, a progressive doping and redistribution of Co 2+ cations diffusing out from the initial cobalt oxide core considering this octahedral-octahedral migration can explain the formation of the mixed ferrites and the final magnetic properties (see schemes of the interfaces between oxides before and after the intermixing for the two samples in figure S6, in the SI). For the CoO@MFO sample, the fact that tetrahedral sites are preferentially occupied by Mn 2+ and Fe 3+ cations in the spinel structure [START_REF] Navrotsky | Thermodynamics of Formation of Simple Spinels[END_REF] reinforces the hypothesis of the Co 2+ cations diffusing out to occupy octahedral sites, offering a prevalent Mn and Co-doped magnetite phase. However, for the CoO@CFO sample, another aspect to take into account is the fact that both CoO-Co3O4 and Co3O4-MFe2O4 interfaces have been reported

to be epitaxially matched at the (111) facets of the fcc sublattice of host O 2-ions, [START_REF] Li | Electrostatic Doping as a Source for Robust Ferromagnetism at the Interface between Antiferromagnetic Cobalt Oxides[END_REF][START_REF] Fontaíña-Troitiño | Exchange Bias Effect in CoO@Fe3O4 Core-Shell Octahedron-Shaped Nanoparticles[END_REF] since the small bulk lattice mismatch between these two lattices can be accomodated by strain. CoFe2O4 was demonstrated to accommodate this epitaxial stress by a continuous migration of Co 2+ from the octahedral to tetrahedral sites, resulting in important changes in the magnetic anisotropy. [START_REF] Ferreiro-Vila | Apparent Auxetic to Non-Auxetic Crossover Driven by Co 2+ Redistribution in CoFe2O4 Thin Films[END_REF] With a redistribution of Co 2+ ions to the tetrahedral sites, we can recalculate the net magnetic moment per formula unit of a spinel ferrite at 0 K, using Néel's two sublattice model. [START_REF] Cullity | Introduction to Magnetic Materials[END_REF][START_REF] Néel | Aimantation à Saturation Des Ferrites Mixtes de Nickel et de Zinc[END_REF] In this model, the net magnetization of the cobalt ferrite with no inversion degree, that is, with the normal spinel structure (Co 2+ [Fe 3+ ]2O4) instead of the inverse one (Fe 3+ [Co 2+ Fe 3+ ]O4), increases up to 7 B in comparison to the 3 B for the inverse structure. Indeed, a favored redistribution of Co 2+ ions to the tetrahedral sites, as reported in thin films, driven by the strain imposed from the cobalt oxide layer below and helped by increasing the temperature, [START_REF] Ferreiro-Vila | Apparent Auxetic to Non-Auxetic Crossover Driven by Co 2+ Redistribution in CoFe2O4 Thin Films[END_REF][START_REF] Sawatzky | Cation Distributions in Octahedral and Tetrahedral Sites of the Ferrimagnetic Spinel CoFe2O4[END_REF] explains the reduced coercivity and the enhanced magnetization measured in the case of the CoO@CFO sample.

Conclusions

In summary, mechanism by which attaining octahedrals of mixed oxides starting from CoO@MFe2O4 (M: Mn, Co) core@shell octahedra is detailed, taking a solid-solid diffusion process into account. This postulated mechanism explains the fact that the hollow/porous nanostructures present a stratified morphology with a magnetically prevailing spinel crystalline structure, on which a progressive doping and redistribution of Co 2+ cations diffusing out from the cobalt oxide initial core determines the final magnetic properties. The fact that tetrahedral sites of the spinel structure are preferentially occupied by Mn 2+ cations forces the Co 2+ cations diffusing out to occupy octahedral sites, offering a prevalent Mn and Co-doped magnetite phase in the CoO-Mn0.4Co1.8Fe0.8O4 sample.

Alternatively, in the CoO-Co1.9Fe1.1O4 sample the Co 2+ cations diffusing out can occupy tetrahedral sites, promoted by the strain generated at the ferrite layers closer to the interfase. In this situation, a normal instead of an inverse cobalt ferrite is attained as the prevalent phase in the final nanostructures, promoting therefore the enhanced magnetization registered. In both cases, the use of interfaces as the starting point in preparative solid-state chemistry is demonstrated, providing an unprecedented degree of control when referred to the magnetic properties of these transition metal oxides at the nanoscale, of which, catalytic, biomedical and energy storage applications, in addition to spintronic applications, can benefit.

Figure 1 .

 1 Figure 1. Lower and higher magnification TEM images with general overviews and corresponding

Figure 2 .

 2 Figure 2. HAADF-STEM images of the nanocrystals from CoO@MFO (framed in blue) and

Figure 3 .

 3 Figure 3. EELS analysis from a nanocrystal of CoO@MFO (top blue panel) or CoO@CFO (bottom

49

 49

Figure 4 .

 4 Figure 4. Representative HAADF tomography images (a and b). 3D tomography reconstruction of

Figure 5 .

 5 Figure 5. Representative HAADF tomography images (a, b and c). 3D tomography reconstruction of

Figure 6 .

 6 Figure 6. HRTEM image (a) and inverse Fourier filtered image using the frequencies that correspond

Figure 7 .

 7 Figure 7. ZFC and FC curves measured applying a 10 mT magnetic field (a, d and g), magnetic

Acknowledgments M. T.-A., B. R.-M. And V. S. acknowledge financial support from the Xunta de Galicia (Regional Government, Spain) under the project ED431C 2016-034 and from the Spanish Ministerio de Economía y Competitividad under project CTM2017-84050-R. M. T.-A. acknowledges financial support from the Xunta de Galicia (Regional Government, Spain) under grant 2017 ED481A. P. T., S. E. and F. P. acknowledge Spanish Government for projects MAT2016-79455-P and Research Network RED2018-102609-T of Ministerio de Ciencia, Innovación y Universidades. The authors thank M. A. Ramos-Docampo for useful discussions about galvanic reactions and R. Mariño-Fernández for the initial synthetic experiments.

hysteresis loops of the three samples registered after ZFC and FC conditions (cooling from 340 K under an applied field of 5 T), in order to obtain information of the interfaces between the different magnetic phases present in the nanocrystals. The FC hysteresis loops of all samples show horizontal and vertical shifts, with a shift along the field axis, defined by HE = (HC+ + HC-)/2, besides an increase of the coercive field (summarized in Table I), all associated to the characteristic exchange bias effect associated to the ferri-antiferromagnetic interface. The initial CoO sample shows this shift in the negative field axis, with 0HE = 24.9 mT, which decreases to 7.6 and 5.5 mT in the case of CoO@MFO and CoO@CFO samples, respectively. While the initial shift in the CoO sample can be attributed to an exchange coupling at the CoO/Co3O4 interface, [START_REF] Li | Electrostatic Doping as a Source for Robust Ferromagnetism at the Interface between Antiferromagnetic Cobalt Oxides[END_REF] the decrease in 0HE registered upon coating can be ascribed to a disruption of the interfaces playing a role. This is indeed more pronounced in the case of the CoO@CFO sample, as evidenced by the larger number of voids within the sample and the more accentuated intermixing process at the new interface established.

Supporting Information

Raman spectra registered from two different spots in the CoO@MFO sample, X-ray powder diffractograms, oxygen K-edge of the EEL spectra recorded in the CoO@MFO sample, derivatives of the ZFC temperature-dependent magnetization and ZFC and FC hysteresis loops measured at 10 K.