Shlomo Dubnov
email: sdubnov@ucsd.edu

Gérard Assayag

Vignesh Gokul
email: vgokul@eng.ucsd.edu

Gerard Assayag
email: gerard.assayag@ircam.fr

Creative Improvised Interaction with Generative Musical Systems

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In our previous studies we developed SOTA generative music models, with focus on interactive machine improvisation that can learn musical style from live or off-line examples and then produce 'more of the same" [START_REF] Assayag | Omax brothers : a dynamic topology of agents for improvization learning[END_REF]. This "same" was interesting in improvisation settings, since the variations maintained resemblance to the immediate expressions of the musician on stage, but were distinct enough to create interest and inspire new interaction. The problem in this setting was that it was the human who found interest in the machine imitation and changed his playing strategies by being inspired by the new machine generated materials, while the artificial agent's generation was oblivious to the musician. In order to allow for more interactivity, special tools were introduced, such as query-based improvisation that biased the choices of the artificial music generator towards materials that had more coherence with the human musician (finding hot spots in the model memory, querymatching, a-priori scenarios and more, see [START_REF] Nika | Dyci2 agents: merging the "free", "reactive", and "scenario-based" music generation paradigms[END_REF] ). Nonetheless, these modifications to the generation policy of the artificial agent that were hard-wired, are largely insufficient to capture the complexity or the subtle expressive inflections in joint multi-musician improvisations. Experiments with multiple artificial musical agents that are capable of listening or influencing each other showed that varying interaction regimes has an important effect on creating interest and prolonging the interaction into a meaningful musical form [START_REF]Creative Symbolic Interaction[END_REF].

The research objective of this inter-disciplinary project is to model and enhance co-creativity as it arises in improvised musical interactions between human and artificial agents, in a spectrum of practices spanning from interacting with software agents to mixed reality involving instrumental physicality and embodiment. Such creative interaction strongly involves co-improvisation, as a mixture of more or less predictable events, reactive and planned behaviors, discovery and action phases, states of volition or idleness. Improvisation is thus at the core of this project and indeed a fundamental constituent of co-creative musicianship, as well as a fascinating anthropological lever to human interactions in general. The outline of the project unfolds as follows:

• Understanding, modelling, implementing music generative and improvised interaction as a general template for symbiotic interaction between humans and digital systems (cyber-human systems)

• Creating the scientific and technological conditions for mixed reality musical systems, based on the interrelation of creative agents and active control in physical systems.

• Achieving distributed co-creativity through complex temporal adaptation of creative agents in live cyberhuman systems, articulated to field experiment in musical social sciences.

This project exploring co-creativity in many dimensions of interaction, learning and generativity is notably linked to the European REACH (Raising Co-creativity in Cyber-human Musicianship) project involving the authors.

On Co-Creativity

The psychologist Margaret Boden has given much attention to the many relations between creativity and machines [START_REF] Boden | Computers models of creativity[END_REF] For her, creativity is the ability to find new, surprising and socially valuable ideas or artifacts, and can occur in three main ways: it can be combinatorial (new configurations of known materials), exploratory (discovering new paths in conceptual / mental spaces) or transformative (when the space itself is disrupted giving way to ideas that were properly inconceivable before). But what is the situation when part of the creativity is delegated to machines, when manifestations of co-creativity emerge from symbolic interactions between human and artificial agents?

In addition to the novelty / effectiveness criteria, cyberhuman co-creativity is strongly felt when two features of improvisation linked to emergence [START_REF] Canonne | A model for collective free improvisation[END_REF] and non-linear dynamics [START_REF] Mouawad | On modeling affect in audio with non-linear symbolic dynamics[END_REF] are identified: (1) emergence of cohesive behaviors that are not reducible to, nor explainable by the mere individual processes of agents; (2) apparition of nonlinear regimes of structure formation, leading to rich musical co-evolution of forms. In our work with jazz improvisers, Bernard Lubat mentions the machine seems, in his words, to "liberate" him, perhaps from specific habits or automatisms. In other words, our inner atlases can be roamed and even modified by creative thinking, in order for the "unthought" (or the yet unthinkable) to find its way.

By producing emergent information structures as a result of cyber-human interaction, we might achieve an epistemological leap [START_REF] Bachelard | La formation de l'esprit scientifique[END_REF] beyond the difficulty of conceding creativity to artificial systems, and assess that creativity is not a state anyway, but rather a dynamical effect of interaction in a complex system, showing radical novelty as a marker of emergence [START_REF] Corning | The re-emergence of "emergence": A venerable concept in search of a theory[END_REF]. By building on this epistemological boost, one would be able to model deep interactions that in turn will trigger co-creative behaviors.

Architecture of Improvising Musical Agents

The architecture of an agent in the improvisation system that we develop is shown in figure 1 The different elements of the system comprise of the following:

• Musical signal : stream of audio or multimedia content This agent architecture can be replicated in a significant number of units in a multi-agent system, producing interaction between artificial agents as well as between agents and humans. Only musical and control signals are exchanged, with multiple cross-feedback loops (when agent A listens to agent B who listens to agent A etc.) that will promote and sustain emergence phases, such as in the Bayesian belief propagation scheme followed in [START_REF] Ken | Probabilistic factor oracles for multidimensional machine improvisation[END_REF]. Signal-symbol quantization [START_REF] Chemla-Romeu-Santos | Cross-modal variational inference for bijective signal-symbol translation[END_REF], constitutes a critical part of the system as the symbolic signal constitute the main vector of information in the internal agent mechanism. Another important part is the capacity for an agent to not only "follow" the musical input from the context, but also to respect user defined scenario, and in more extreme cases, to incrementally generate scenarios by itself as in [START_REF] Carsault | Combining real-time extraction and prediction of musical chord progressions for creative applications[END_REF], where a LSTM with bottleneck

Reduced Representation

Studies of human cognition suggest Rate-Distortion as a way of extracting useful or meaningful information from noisy signals [START_REF] Sims | Rate-distortion theory and human perception[END_REF]. The idea of reduced representation also has been recently explored in the context of representation learning in deep neural networks using a framework known as Information Bottleneck [START_REF] Tishby | Deep learning and the information bottleneck principle[END_REF]. In deep learning some attempts to consider predictive information through use of a bottleneck or noisy representation in temporal models such as RNNs have recently appeared in the literature [START_REF] Alemi | Variational predictive information bottleneck[END_REF], [START_REF] Dong | On predictive information in rnns[END_REF]. Accordingly, in order to achieve a better interaction between the human and the machine, we are seeking two types of data reduction:

• lossy representation of the signal (audio or midi) that effectively reduce the dimensionality of the latent representation and allow for better generalization

• symbolization of the lossy encoding to allow for better temporal representation by using language modeling with variable memory length Learning music representation with auto-encoder, a schematic representation of the noise induced by bitreduction is given by Figure 2. Performing finite bit-size encoding and transmission of the quantized latent values from encoder Z e to decoder Z d is not required, since we are interested in gating and biasing the original signal towards the prior distribution by encoding it at a limited bit-rate, which is given by the following optimal channel [START_REF] Berger | Rate distortion theory; a mathematical basis for data compression[END_REF] To illustrate the effect of reduced data representation on predictive properties of music, we performed quantization at different rate for a monophonic encoding of music using a disentangled VAE training for a dataset of 14 two-part inventions composed by Johann Sebastian Bach. The MIDI files are collected from the Complete Bach MIDI Index1 . Mutual information neural estimation (MINE) [START_REF] Belghazi | Mutual information neural estimation[END_REF] was used to analyze the relations in time and across voices from their reduced latent representations.

Q(z d |z e ) = N ormal(µ d , σ 2 d ) (1)
µ d = z e + 2 -2R (µ e -z e ) (2)
σ 2 d = 2 -4R (2 2R -1)σ 2 e (3)
From the results, we conclude that the mutual information values between conditional latent variables and predictive latent variables depend on the level of reduced representation. We find that reducing bit-allocation can effectively improve the mutual information between conditional latent variables and predictive latent variables for each scenario. For more details of this and other polyphonic and audio experiments we refer the readers to [START_REF] Dubnov | Deep music information dynamics: Novel framework for reduced neuralnetwork music representation with applications to midi and audio analysis and improvisation[END_REF].

Musical Information Dynamics

Assuming the music signal X = x[n] is encoded into a sequence of latent representations Z = z[n], with n denoting discrete time step n. We would like to algorithmically discover the sequential structure of Z, and be able to present the structures quantitatively. Music Information Dynamics (MID) [START_REF] Abdallah | Information dynamics: Patterns of expectation and surprise in the perception of music[END_REF][START_REF] Dubnov | Musical information dynamics as models of auditory anticipation[END_REF][START_REF] Pearce | Statistical learning and probabilistic prediction in music cognition: mechanisms of stylistic enculturation[END_REF] provides a theoretical framework that utilizes mutual information between past and present observations to model the predictability of the signal. The advantage of adopting MID is that it optimizes or calculates an information theoretic measurements on the input sequence Z and is agnostic of specific sequence related applications, such as motifs discovery or structure segmentation. MID was shown to be important for understanding human perception of music in terms of anticipation and predictability [START_REF] Abdallah | Information dynamics: Patterns of expectation and surprise in the perception of music[END_REF][START_REF] Dubnov | Musical information dynamics as models of auditory anticipation[END_REF].

An efficient formal method for studying MID for sequence Z[n] is the Information Rate (IR) that considers the relation between the present measurement Z = z[n] and

I(Q(Z), Q ←- (Z)) (4) = H(S) -H(S| ← - S ) (5)
According to this measure, the maximal value of IR is obtained when the difference between the uncertainty of H(S) and predictability H(S| ← -S ) is at its greatest, meaning that there is a balance between variation and predictability. Quantization Q(Z) is needed due to the need to detect inexact repetitions in the sequence Z, which in turn signifies the allowed level of similarities between observations in Z, or the amount of signal detail that is significant when comparing the present to the past.

Symbolization and Music Analysis using VMO

Variable Markov Oracle (VMO) [START_REF] Wang | Guided music synthesis with variable markov oracle[END_REF] accepts a representation Z = z [START_REF] Abdallah | Information dynamics: Patterns of expectation and surprise in the perception of music[END_REF], z [START_REF] Alemi | Variational predictive information bottleneck[END_REF], . . . , z[N ] and turns it into a symbolic sequence S = s [START_REF] Abdallah | Information dynamics: Patterns of expectation and surprise in the perception of music[END_REF], s [START_REF] Alemi | Variational predictive information bottleneck[END_REF], . . . , s[n], . . . , s[N ], with M states over a finite alphabet Σ. The labels are formed by finding suffixes in a graph structure constructed by the VMO algorithm. Due to space consideration, we leave out the VMO construction and refer the readers to [START_REF] Dubnov | Audio oracle analysis of musical information rate[END_REF][START_REF] Wang | Music pattern discovery with variable markov oracle: A unified approach to symbolic and audio representations[END_REF]. The essential step in symbolization is finding a threshold with the highest MID value. The threshold θ partitions the space of features into categories that capture and represent the different sound elements by determining if the incoming z[n] is similar to one of the frames following previous instances in the sequence pointed to be a suffix link from n -1. VMO symbolization step assigns two frames z[i] and z[j] the same label s

[i] = s[j] ∈ Σ if ||z[i] -z[j]|| ≤ θ.
To find the optimal threshold θ, MID measure can be estimated by any predictive compression algorithm C(•). The compression gain over blocks of symbols is used to replace the the entropy term H(•) as our measure of complexity [START_REF] Lefebvre | Compror: on-line lossless data compression with a factor oracle[END_REF]

IR(Z) = max θ,s[n]∈Σ θ [C(s[n]) -C(s[n]| ← - S )]. (6)
It should be noted that the alphabet out of the quantization is constructed dynamically, as new labels can be added when an input sample cannot be assigned to one of the existing clusters of samples already labeled by existing labels.

As an example of the effect of different levels of symbolization on discovery of the motif structure in different musical styles, we performed comparative analysis of several works for the flute [START_REF] Dubnov | Towards cross-cultural analysis using music information dynamics[END_REF] using human engineered (Chroma and MFCC) and machine learned representations (VAE). We provide a partial example of the finding in the figure 3. It can be seen that the Dongxiao music is characterized by much shorter motifs, which were found at much finer threshold value compared to Telemann that is characterized by longer motifs that required a coarse quantization. In a different work, a VMO based MID estimator was used to evaluate the performance of generative recurrent latent models for MIDI data. The results showed that Variational encoding, which added randomization into the latent representation of the generative model, resulted in an improved motif structure of musical generation output as it better resembled the motif statistics found in the original data compared to RNN methods that tend to overly reproduce repetitions having looping sub-sequences [START_REF] Koh | Rethinking recurrent latent variable model for music composition[END_REF].

RL tuning of musical generators

There have been a lot of research in modelling music generation and synthesis as a Maximum Likelihood Estimation (MLE) problem [START_REF] Zhu | Xiaoice band: A melody and arrangement generation framework for pop music[END_REF][START_REF] Dhariwal | Jukebox: A generative model for music[END_REF][START_REF] Huang | Music transformer[END_REF], where the objective is to predict the next best likely note/symbol. However, one of the key issues is exposure bias -the training and testing conditions might be completely different, due to the vast possibilities in music improvisation. Hence, we need a mechanism to facilitate zero-shot learning. Another issue is Greedy vs Dynamic Programming -in MLE, the approach is greedy i.e the note/symbol which is best suited for the current time-step is chosen. This might cause issues in longterm music generation, as we need a solution to optimize for long term rewards (dynamic programming). We solve the above issues by treating music improvisation/generation as a reinforcement learning problem. An agent maximizes a long-term discounted reward function by exploring different states. In order to achieve this, some key challenges in applying RL to music need to be addressed:

• Representation of States: a trivial solutions could be representing a state as the sequence of notes played so far, or sequence of past spectral feature, symbolized into a discrete representation. This poses a problem as we need a state space independent of the previous observations, so as to allow the musical agent to explore. Recent works [START_REF] Dhariwal | Jukebox: A generative model for music[END_REF] have used similar representations to model raw audio. They use MLE to synthesize audio sequentially, by fitting another model, say a transformer, on top of these learned representations. This is extremely time consuming and also computationally expensive. Since, our goal is to have real-time music improvisation, we need to parallelize this synthesis process.

We address the problem of representation by considering VMO symbolization and using ideas from VQ-VAE [START_REF] Van Den | Neural discrete representation learning[END_REF] and VQ-GAN [START_REF] Iashin | Taming visually guided sound generation[END_REF]. We use raw audio signals as the training dataset to train a codebook model to obtain the vector-quantized embeddings (codified audio) as our representation. The representation of a state would be the combined sequences of vector-quantized embeddings of both the user input audio and the machine improvised audio observed so far. More formally, given a raw audio signal (user input) x and a VQ model E, we represent the signal as

E(x) = [h 1 , h 2 , h 3 , h 4 , ..., h n ],
where h 1 ,h 2 ,h n are indices corresponding to the embedding vectors in the codebook. Similarly, we represent the generated music signal as a sequence of codes [m 1 , m 2 , ..., m n ]. To represent a state s t , we use a concatenated representation of [s ht , s mt ] where s ht = h 0:t-1 and s mt = m 0:t-1 .

We design our rewards based on recent works on using RL for music synthesis [START_REF] Jiang | Rl-duet: Online music accompaniment generation using deep reinforcement learning[END_REF][START_REF] Jaques | Tuning recurrent neural networks with reinforcement learning[END_REF]. Given state s t at timestep t, our reward agent assigns a score based on p(m t-∆t : m t+∆t |s t ), where m t denotes the codified audio generated by our synthesizer at timestep t and ∆t is a parameter to control the window size. Note that this is different than MLE, as the RL agent tries to maximize the discounted For negative reinforcement, the reward agent penalizes compositions, where the pitch remains the same for more than n timesteps. Our framework also allows for customization where the rules of different synthesizers can be customized according to the needs of the user. The reward agent for the composer is based on the mutual information between the observed input signal and the current signal of the music agent.

For our synthesis component, we choose a RL module, where each action is replaced with skills (options) [START_REF] Barreto | The option keyboard: Combining skills in reinforcement learning[END_REF]. We can think about these skills as a series of temporally extended courses of actions. Our RL agent consists of two components: lower-level components (specializers) that are specialized in a skill, and a higher level component (composer) that synthesizes improvisations by using a combination of the lower level components. Each of the specializers are generators that output sequences of vector-quantized codes such that the decoded raw audio satisfies a particular rule, i.e optimizes for a reward function. The composer is responsible for selection of options or combining these options to compose improvisations to human-played compositions at real-time.

Figure 1 .

 1 Figure 1. The flow of information between listen, learn and interact modules and the diverse data and control feed-backs channels in an agent

Figure 2 .

 2 Figure 2. Noisy channel between encoder and decoder

  it's past ← -Z = z[1], z[2], . . . , z[n], . . . , z[N ], formally defined as the maximum of mutual information over different quantized level of the sequence S = Q(Z) IR(Z) = max Q:S=Q(Z)

Figure 3 .

 3 Figure 3. Motifs found in different flute pieces using the best VMO for VAE self , Chroma features, and MFCCs.

Figure 4 .

 4 Figure 4. Percussion phrase generated with CNN trained on music notation, without (top) and with (bottom) RL refinement

Table 1 . The experiment results of control- ling the rate to measure the mutual informa- tion between the conditional latent variables and the predictive latent variables. The first column shows different predictive scenarios. The left columns show the mutual information with different rates.

 1

	Scenario	R=10	R=100 R=1000 R=10000 Original
	past-future	67.142 132.422 73.089	83.012	75.120
	1st voice-2nd voice. 36.963 148.054 93.637	77.848	126.631
	2nd voice-1st voice 61.643 91.893 104.920	91.821	82.037

•

  Reward function: Another key aspect of RL is finding a good reward function. The rewards are responsible for tuning the agent towards 'good behavior'. In an open-ended topic such as music improvisation, it can be very hard to quantify what 'good music' is across different styles and cultures.

http://www.bachcentral.com/midiindexcomplete.html


Dafan Du

Guang Guan

Annie Gagnoud

Yves Fautrelle

Zhongming Ren

Xionggang Lu

Hui Wang

Yinming Dai

Qiuliang Wang

Xi Li

Xili

Effect of a high magnetic field on the growth of ϵ-CuZn5 dendrite during directionally solidified Zn-rich Zn-Cu alloys

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Zn-Cu is a typical binary peritectic system in which four peritectic reactions occur over the whole composition range. Peritectic reactions are commonly observed in systems that include components with large differences in melting points [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF]. It not only features in the solidification of structural materials [START_REF] Uhlmann | [END_REF][3][4][5] but also in the production of superconductors such as YBCO [6,7] and magnetic materials such as Co-Sm-Cu and Nd-Fe-B [8,9]. The dendritic structure is common during directional solidification of peritectic alloys. Recently, due to the development of superconducting magnets, a high magnetic field is widely used to improve material properties during material processing [10]. The effect of a high magnetic field on the dendrite growth in single phase alloys has been studied, and the results indicated that the application of a high magnetic field significantly affected the dendrite array significantly [11]. However, a few works have been done to study the effect of a high magnetic field on the dendrite growth in peritectic alloys. In present work, the effect of a high magnetic field on the growth of the ε-CuZn 5 phases in the Zn-rich Zn-Cu peritectic alloys has been investigated. Moreover, the Seebeck thermoelectric signal (E S ) at the liquid/ solid interface in the Zn-5 wt.% Cu peritectic alloy was measured insitu and the results indicated that the value of the Seebeck thermoelectric force was on the order of 10 μV.

The experimental results indicated that the application of the magnetic field had modified the dendrite morphology in the Zn-rich Zn-Cu alloys and changed the crystalline orientation. The above experiment phenomena were analyzed from the magnetic crystalline anisotropy and thermoelectric magnetic effects. This work will probably yield basic and valuable information leading to an extension of magnetic field application to exert microstructure and property control on metallic materials.

Experimental

The Zn-Cu alloys used in this study were prepared with high purity Zn (99.999 wt.%) and Cu (99.999 wt.%) in an induction furnace. The alloy, being put in a high-purity graphite crucible of 10 cm diameter, was heated to 600 °C, magnetically stirred for 0.5 h and poured into a graphite mold to cast specimens with the diameter of 3 mm and length of 200 mm. The cast specimen was enveloped in the tube of high purity corundum. The inner diameter of the sample is 3 mm and its length is 200 mm. The experimental apparatus, which is shown in Fig. 1,consists of a static superconductor magnet and a Bridgman-Stockbarger type furnace equipped with pulling system and temperature controllers. The superconductor magnet can produce an axial static magnetic field with an adjustable intensity up to 14 T. The furnace, which consists of non-magnetic material, has a negligible effect on the field uniformity. A water-cooled cylinder containing liquid Ga-In-Sn metal (LMC) was used to cool down the specimen. The temperature gradient in the specimen was controlled by adjusting the temperature of the furnace hot zone, which is insulated from the LMC by a refractory disc. The apparatus was designed such that the specimen moves downward while the furnace remains stationary. The growth speeds, R, lies in the range 0.5 μm/s ≤ R ≤ 15,000 μm/s. To observe the morphology of the liquid/ solid interface during the growth of the crystal, quenching experiments were performed by quickly withdrawing the specimen into the LMC cylinder to cool the specimen immediately to room temperature. During the experiment the specimens contained in the corundum crucibles were melted and directionally solidified in the Bridgman apparatus by drawing the crucible assembly at various speeds into the LMC cylinder. In all of the experimental procedures, the magnetic field was present. The samples obtained from the experiments were extracted from both longitudinal and transversal sections of the directionally solidified casting. The samples were further ground with silicon carbide papers up to 2400 mesh, polished and etched with the Palmerton's reagent (40 g CrO 3 ;1.5gNa 2 SO 4 and 200 ml of distilled and deionized water) to reveal the microstructure. Microstructural characterization of specimens was observed with a Leica DM 6000M optical microscope. Crystallographic characteristics were measured through the EBSD using a fieldemission scanning electron microscope (SEM, SU-70, HITACHI) equipped with the EDAX's Hikira XP2 camera and OIM Analysis System (EBSD, EDAX Inc.). For EBSD analysis, the samples were first polished using a glycol based SiC suspension and then final vibration polishing using 50 nm colloidal silica on a vibratory polisher (Buehler, model: Vibromet 2) for 1.5 h. Standard clean-up procedures [START_REF] Oim | User's Manual[END_REF] was performed in OIM study to gain high-quality EBSD figures, as follows: (i) grain dilatation with a grain tolerance angle (GTA) of 2°; a minimum grain size (MGS) of three pixels; (ii) neighbor CI correlation with minimum CI of 0.1.

Results

Fig. 2 shows the longitudinal structures for directionally solidified Zn-5 wt.% Cu alloy at various growth speeds without and with a 10 T magnetic field. It is obviously seen that solidified structures consist of primary ε-CuZn 5 phase (white) and η-Zn phase (dark) below the solid-liquid interface in each samples. It can be observed that the dendrite morphology without the magnetic field is typical columnar structure. However, when a 10 T high magnetic field was applied, regular columnar dendrites were destroyed significantly and tended to transform into equiaxed grains. Moreover, longitudinal microstructures of directionally solidified Zn-3.5 wt.% Cu alloy at a growth speed of 5 μm/s under various magnetic fields have been investigated and the results are shown in Fig. 3. It can be learned that with the application of a 1Tmagneticfield, the regular dendrites were broken and irregular dendrites formed. Continuing to increase the magnetic field to 5 T and 10 T, the columnar-to-equiaxed transition (CET) occurred. Fig. 4 shows the longitudinal microstructures near the liquid/solid interface and transverse structure 5 mm below the interface in directionally solidified Zn-5.0 wt.% Cu alloy at a growth speed of 10 μm/s without and with a 10 T magnetic field. It can be observed that with the absence of the magnetic field the ε-CuZn 5 dendrite exhibits a typical columnar dendrite microstructure as shown in Fig. 4a andc and the typical primary ε-CuZn 5 phase uniform distribute across the entire sample cross section. However, when a 10 T high magnetic field is imposed, the microstructures (Fig. 4b andd) are changed significantly; the ε-CuZn 5 columnar dendrite was destroyed seriously and transformed into irregular equiaxed grains.

Further, the EBSD technology was applied to study the dendrite morphology and the crystalline orientation. Fig. 5 shows the EBSD map for the structures in directionally solidified at low growth speeds (2 μm/s and 10 μm/s) without and with a 10 T magnetic field. One can notice that the magnetic field caused the deformation of the columnar η-Zn and ε-CuZn 5 phases and the occurrence of the CET. As the growth and orientation behavior of the ε-CuZn 5 phases in the Zn-Cu alloy are unclear so far, firstly the dendrite growth and orientation behavior of the ε-CuZn 5 phases was studied by using the EBSD analysis. Fig. 6 and Fig. 7 show the EBSD map and the corresponding b 0001N-pole figures for the structures in directionally solidified Zn-5 wt.% Cu alloy at various growth speeds with the absence of magnetic field, respectively. One can notice that the ε-CuZn 5 phases exhibit irregular columnar structure at low growth speeds and the regular dendrites formed at a moderate growth speed. The pole figures in Fig. 7a-eshows that the ε-CuZn 5 dendrite possessed a very strong fiber texture with b 0001N-directions of the hexagonal ε-CuZn 5 crystals oriented preferentially parallel to the growth direction and the results also show that b 0001 N-directions of the hexagonal ε-CuZn 5 crystals oriented along the growth direction gradually with the increase of growth speed. However, when the growth speed increases to 100 μm/s, the b 0001 N fiber texture disappeared and the b0001 N-direction deflect from the growth direction (i.e. magnetic field direction). Fig. 8 shows the EBSD map and the corresponding b 0001 N-pole figures of the ε-CuZn 5 phase for the longitudinal structures in directionally solidified Zn-5 wt.% Cu alloy at various growth speeds under a 10 T magnetic field. It can be observed clearly that the application of a high magnetic field caused the regular columnar dendrite to transform into equiaxed grains. At the same time, the b0001 N-preferred growth direction of the ε-CuZn 5 phases deflected from the solidification direction and became disperse. This implies that the columnar dendrites orienting with the b0001N-crystal direction along the solidification direction have been transformed into equiaxed grains with various random orientations under the magnetic field. Further, Fig. 9 shows the EBSD analysis of the transverse section for samples solidified without and with a 10 T magnetic field at various growth rates.

Discussion

Above experimental results revealed that the application of a high magnetic field during directional solidification caused the formation of dendrite fragments and the occurrence of the CET. Normally, two major magnetic effects will influence the dendrite growth during the directional solidification under the magnetic field. One effect is the magnetic moment arising from the magnetic crystalline anisotropy. This moment will tend to align the dendrites with the easy magnetic axis along the magnetic field [START_REF] Fa | [END_REF]. Thus, the primary arm of the dendrite tends to deviate from the solidification direction as the easy magnetic axis rotates towards the magnetic field direction. As a consequence, the dendrite array is destroyed. Moreover, owing to the Seebeck effect [15], a thermoelectric current will be generated on the tip of the cell/ dendrite [16]. When the magnetic field is applied, the interaction between the current and the magnetic field will induce a thermoelectric magnetic force. This thermoelectric magnetic force also destroys the dendrite array. Above two magnetic effects may work simultaneously and modify the dendrite array during directional solidification in the magnetic field.

Firstly, the magnetic crystalline anisotropy of the ε-CuZn 5 crystal is studied. To study the magnetic crystalline anisotropy of the ε-CuZn 5 crystal, the alignment behavior of the ε-CuZn 5 primary phase under a magnetic field was investigated. Fig. 10 shows the EBSD map and the corresponding b0001N pole figures for the microstructures during the diffusion process in the Zn/Cu diffusion couple with a 16 T magnetic field. The detail experimental process and setup for the diffusion experiment are available in Ref. [17]. It can be seen that the regular CuZn 5 hexagons appeared on the transverse section of the sample fabricated with the magnetic field and is oriented with the b0001 N-crystal direction along the magnetic field direction. Therefore, the b0001 Ncrystal of the CuZn 5 crystals is not only an easy magnetization axis but also a preferred growth direction. Thus, the magnetic moment becomes zero during the directional solidification under an axial magnetic field, and the effect of the magnetic moment on the dendrite array is eliminated. The modification of the microstructure during directional solidification under the magnetic field should be attributed to the thermoelectric magnetic effects.

Furthermore, to examine the distribution and amplitude of the thermoelectric magnetic effects near the liquid/solid interface during directional solidification of the Zn-rich Zn-Cu alloys under an axial magnetic field, numerical simulations were performed using the finite element commercial code FLUX-EXPERT™, which is available in our laboratory (SIMAP-EPM/G-INP/CNRS). The study configuration is 2 D, and the liquid and solid regions are both considered. The physical properties used in the computations are listed in Table 1 and thermoelectric power difference (η sl = S S -S L ) of the Zn-5 wt.% Cu alloy near the liquid/solid interface was measured in-situ by MEPHISTO apparatus for simulation. The basic principle of the interfacial thermoelectric measurement using the Seebeck technique was schematically presented in Fig. 11a. The details of the principle had been discussed in Refs. [19,20]. Fig. 11b shows the Seebeck response obtained for the Zn-5 wt.% Cu alloy at a growth speed of 5 and 20 μm/s, respectively. One can learn that a plateau is obtained on the Seebeck curve to indicate establishment of steady state, and then the signal drops to a new steady state when furnace translation is shut off. The magnitude of signal drop, E S , corresponds directly to the effective liquid/solid thermoelectric power difference (η sl ). This indicates that there indeed exists a liquid/solid thermoelectric power difference during directional solidification of the Zn-5 wt.% Cu alloy. The results show that there indeed exists a liquid/ solid thermoelectric power difference during directional solidification of the Zn-5 wt.% Cu alloy and the Seebeck signal at the given condition is on the order of 10 μV.

Fig. 12 shows the mesh used for the numerical simulation, the typical distribution of current density and thermoelectric magnetic force around a modeled liquid/solid interface under a 0.5 T magnetic field.

The current swirl appears in the vicinity of the interface and that the rotation of the liquid around the dendrite, due to the radial component of the current density, will produce (see Fig. 12b). It should be noted that this swirl is remarkably localized in the vicinity of the interface and not extend into the bulk of the liquid. Thus, when a magnetic field is imposed, the interaction between the magnetic field and the thermoelectric current will produce a thermoelectric magnetic force as shown in Fig. 12c. The value of the force acting on the solid increases linearly as the magnetic field intensity increases (see Fig. 12d). Here, it is important to note that the thermoelectric magnetic force imposed on the solid during directional solidification with a temperature gradient of 60 K/cm under a 10 T magnetic field is of the order of 10 7 N/m 3 . The thermoelectric magnetic force acting on the solid may induce a torque that disrupts the dendrite resulting in the formation of dendrite fragments. As a consequence, the number of equiaxed grains ahead of the columnar front increases.

According to Hunt's mode, when the volume fraction of equiaxed grains exceeds 0.49, the CET will occur [18].

Conclusions

High magnetic fields were applied to study the growth of ε-CuZn 5 columnar dendrite for the peritectic Zn-Cu alloys. The results reveal that the high magnetic fields change the growth behavior of ε-CuZn 5 dendrite effectively.

1. The regular ε-CuZn 5 columnar dendrite formed and was oriented with the b 0001N-crystal direction along the solidification direction at moderate growth speeds. The application of a high magnetic field caused the deformation of the ε-CuZn 5 phases and the occurrence of the CET of the ε-CuZn 5 phases. 2. The ε-CuZn 5 crystal owns a magnetic crystalline anisotropy and the b0001 N-crystal direction is its easy magnetization axis. Indeed, it was found that the regular CuZn 5 hexagons appeared on the transverse section of the sample fabricated with the magnetic field and was oriented with the b0001N-crystal direction along the magnetic field direction. 3. Numerical analysis revealed that a thermoelectric magnetic force acted on the solid and the value of this force increased linearly as the magnetic field intensity increased. The breakup and the CET of the ε-CuZn 5 phases during directional solidification under the magnetic field may be primarily attributed to the thermoelectric magnetic force acting on the solid.

Table 1

Physical properties of Zn-Cu alloys [START_REF] Huebener | Thermoelectricity in metals and alloys[END_REF][22][23].

Properties Magnitude

Electrical conductivity of solid Zn (σ S1 , Ω -1 m -1 ) 16.6 × 10 6 Electrical conductivity of solid CuZn 5 (σ S2 , Ω -1 m -1 ) 15.8 × 10 6 Electrical conductivity of liquid (σ L , Ω -1 m -1 ) 8.8 × 10 -6 Temperature gradient (G, K m -1 )6 0 × 1 0 2 Kinematic viscosity (ν,ms -1 ) 1.2 × 10 -7 Density (ρ,kgm -3 ) 7.14 × 10 3 (Nos. 51271109, 51171106 and 51225401) and the Science and Technology Committee of Shanghai (No. 14JC1491400).

Fig. 1 .

 1 Fig. 1. Schematic illustration of the Bridgman solidification apparatus in a superconductor magnet.

Fig. 2 .

 2 Fig. 2. Longitudinal microstructures in directionally solidified Zn-5 wt.% Cu alloy at various growth speeds without and with a 10 T magnetic field: (a) 5 μm/s; (b) 10 μm/s; (c) 50 μm/s; and (d) 100 μm/s.

Fig. 3 .

 3 Fig. 3. Longitudinal structures in directionally solidified Zn-3.5 wt.% Cu alloy at a growth speed of 5 μm/s under various magnetic fields: (a) 0 T; (b) 1 T; (c) 5 T; and (d) 10 T.

Fig. 4 .

 4 Fig. 4. Microstructures near the liquid/solid interface in directionally solidified Zn-5 wt.% Cu alloy at a growth speed of 10 μm/s without and with a 10 T magnetic field: (a) longitudinal structure, 0 T; (b) longitudinal structure, 10 T; (c) transverse structure, 0 T; and (d) transverse structure, 10 T.

Fig. 5 .

 5 Fig. 5. EBSD map for the longitudinal structures in directionally solidified Zn-5 wt.% Cu alloys at a certain growth speed with and without a 10 T magnetic field: (a) 2 μm/sand(b)10μm/s.

Fig. 6 .

 6 Fig. 6. EBSD map of the ε-CuZn 5 phase on the longitudinal structures in directionally solidified Zn-5 wt.% Cu alloy at various growth speeds: (a) 2 μm/s; (b) 5 μm/s; (c) 10 μm/s; (d) 20 μm/ s; (e) 50 μm/s; and (f) 100 μm/s.

Fig. 7 .

 7 Fig. 7. The b0001N-pole figures of the ε-CuZn 5 phase corresponding to the EBSD map in Fig. 6: (a) 2 μm/s; (b) 5 μm/s; (c) 10 μm/s; (d) 20 μm/s; (e) 50 μm/s; and (f) 100 μm/s.

Fig. 8 .Fig. 9 .

 89 Fig. 8. EBSD map and the corresponding b0001N-pole figures of the ε-CuZn 5 phase for the longitudinal structures in directionally solidified Zn-5 wt.% Cu alloy at various growth speeds under a 10 T magnetic field: (a) 10 μm/s; (b) 20 μm/s; and (c) 100 μm/s.

Fig. 10 .

 10 Fig. 10. EBSD map and the corresponding b0001N-pole figures for the structures annealed at 723 K for 7 h with a 16 T magnetic field: (a) EBSD map; (b) IPF map for ε-CuZn 5 phase in (a); (c) and (d) showing b0001N-pole figures of the Zn and CuZn 5 phases corresponding to (a), respectively.

Fig. 12 .

 12 Fig. 12. Computed thermoelectric magnetic effects around the modeled liquid/solid interface during directional solidification of the Zn-Cu alloys under a 0.5 T magnetic field: (a) Mesh for the numerical simulation; (b) thermoelectric current streamlines near the interface (in A/m); (c) thermoelectric magnetic force in liquid and solid near the liquid/solid interface (in N/m 3 ); and (d) value of the thermoelectric magnetic force acting on the solid as a function of the magnetic field intensity.

Fig. 11 .

 11 Fig. 11. (a) Schematic diagram of the MEPHISTO apparatus and (b) Seebeck response obtained for Zn-5 wt.% Cu alloy at various growth speed.

Acknowledgments

This work is supported by the National 973 Project (Nos. 2014CB643403 and 2011CB610404), the European Space Agency through the IMPRESS project, Natural Science Foundation of China