Frustrated magnets realize exotic forms of quantum matter beyond conventional order. Due to a lack of controlled and unbiased methods to study frustration in three dimensions, many questions remain unanswered. While most established numerical techniques have limited applicability, approaches based on cluster expansions are promising alternatives. By design, they do not suffer from dimensionality or frustration and generate reliable insights into the thermodynamic limit without any restriction in the parameter space. This thesis makes significant methodological progress in controlled numerical approaches tailored to study frustration in three dimensions. It covers (i) an automatic detection algorithm for symmetries in generic clusters, (ii) a general approach to the numerical linked cluster algorithm to study finite - and zero - temperature properties, and (iii) an expansion method based on the linked cluster theorem to obtain a suitable dressing for valence-bond crystals. In particular, we study one of the archetypal problems of frustrated magnetism in three dimensions: the pyrochlore Heisenberg antiferromagnet. For the first time, we are able to unbiasedly resolve its thermodynamic quantities to a temperature far beyond the scale on which the Schottky anomaly occurs. The broad applicability of the numerical linked cluster algorithm allows for the systematic investigation of different spin-liquid candidate materials such as the Cerium-based pyrochlores Ce₂Zr₂O₇ and Ce₂Sn₂O₇. Despite a similar chemical composition, the algorithm finds fundamental differences in their quantum mechanical nature by constraining their microscopic exchange parameters. Zero temperature properties are even less accessible: Neither the nature of the ground state nor an estimate of its energy are known for the pyrochlore antiferromagnet. Large-scale density matrix renormalization group calculations pushed to three dimensions provide the first reliable estimate of its ground-state energy and yield robust evidence for a spontaneous inversion symmetry breaking manifesting itself as an energy density difference on the tetrahedral sublattice. The symmetry-breaking tendency of the model is further observed in the presence of an external magnetic field where similar calculations suggest a stable 1/2-magnetization plateau. Continuing the investigation of low-energy states, we propose a new family - exponentially numerous in the linear system size - of valence-bond crystals as potential ground states. Understanding the stability of the previously overlooked family of states suggests a remarkable change of perspective on frustration with a focus on unfrustrated motifs. In sum, these discoveries present significant progress towards resolving long-standing questions regarding the nature of the ground state of the quantum pyrochlore S=1/2 antiferromagnet. / Frustrierte Magnete realisieren exotische Formen von Quantenmaterie, welche gewöhnliche Ordnungen übersteigen. Viele etablierte numerische Methoden versagen bei Frustration in drei Dimensionen, da sie entweder nicht anwendbar sind, unkontrolliert sind oder bestimmte Zustände vorziehen. Clusteralgorithmen bilden eine vielversprechende Alternative. Sie erfahren keine Einschränkung durch die Dimensionalität oder die Frustration des Problems und erlauben daher zuverlässige Einblicke in den thermodynamischen Limes. Diese Arbeit präsentiert methodische Fortschritte von kontrollierbaren Ansätzen, welche auf frustrierte Systeme in drei Dimensionen zugeschnitten sind. Sie beinhaltet (i) die Entwicklung eines Algorithmus zur automatischen Detektion räumlicher Symmetrien für allgemeine Cluster, (ii) einen allgemeinen Zugang zum 'numerical linked cluster algorithm'', um Eigenschaften bei endlicher Temperatur und dem absoluten Nullpunkt zu studieren und (iii) einen Clusteralgorithmus zur Optimierung des Zustands eines 'valence-bond' Kristalls. Die methodischen Fortschritte dieser Arbeit tragen zur Lösung eines archetypischen Problems von frustriertem Magnetismus in drei Dimensionen bei: dem Pyrochlor Heisenberg Antiferromagnet. Sie erlauben zuverlässige Einblicke in die Thermodynamik bis hin zu nicht-trivialen Temperaturen weit unter der Schottky-Anomalie. Die weiten Anwendungsmöglichkeiten des Clusteralgorithmus macht die systematische Untersuchung von Spinflüssigkeitskandidaten, wie die auf Cer basierenden Pyrochlore Ce₂Zr₂O₇ und Ce₂Sn₂O₇, möglich. Trotz einer ähnlichen chemischen Komposition, findet der Algorithmus fundamentale Unterschiede in ihrer quantenmechanischen Struktur. Frustration in drei Dimensionen ist am absoluten Nullpunkt ähnlich unzugänglich wie bei endlicher Temperatur und weder der Grundzustand, noch Schätzungen der Grundzustandsenergie des Pyrochlor Antiferromagneten sind bekannt. Groß angelegte Dichtematrixrenomierungsgruppenrechnungen in drei Dimensionen ermöglichen erstmals eine verlässliche Schätzung der Energie und finden eine spontan gebrochene Inversionssymmetrie, welche durch einen Unterschied in der Energiedichte auf dem tetraedrischen Untergitter ausgedrückt ist. Die Tendenz, die Symmetrie des Systems zu brechen, ist auch in der Präsenz eines externen magnetischen Feldes zu beobachten. Rechnungen deuten die Stabilität des 1/2-Magnetisierungsplateaus an. Einen signifikanten Beitrag zum Verständnis des Heisenberg-Models auf dem Pyrochlor wird durch eine Familie von potentiellen Grundzuständen geleistet, welche als harte Hexagone im Gitter visualisiert werden können. Ihre Anzahl skaliert exponentiell in der linearen Systemgröße und ihre niedrige Energie eröffnet eine neue Sichtweise auf frustrierte Magnete, welche den Fokus auf unfrustrierte Geometrien lenkt. Im Widerspruch zu der prominenten Spinflüssigkeitsannahme deuten die Ergebnisse dieser Arbeit auf Ordnung im Pyrochlor Antiferromagneten hin.

Mining will remain indispensable for the foreseeable future. For millennia, our society has been exploring and exploiting mineral deposits. Consequently, most of the easily exploitable high-grade deposits, which were of primary interest given their obvious technical and economic advantages, have already been depleted. For the future, the mining sector will have to efficiently produce metals and minerals from low-grade orebodies with complex mineralogical and microstructural properties -- these are generally referred to as complex orebodies. The exploitation of such complex orebodies carries significant technical risks. However, these risks may be reduced by applying modelling tools that are reliable and robust. In a broad sense, modelling techniques are already applied to estimate the resources and reserves contained in a deposit, and to evaluate the potential recovery (i.e., behaviour in comminution and separation processes) of these materials. This thesis focusses on the modelling of recovery processes, more specifically mineral separation processes, suited to complex ores. Despite recent developments in the fields of process mineralogy and geometallurgy, current mineral separation modelling methods do not fully incorporate the available information on ore complexity. While it is well known that the mineralogical and microstructural properties of individual particles control their process behaviour, currently widely applied modelling methods consider only distributions of bulk particle properties, which oftentimes require much simplification of the particle data available. Moreover, many of the methods used in industrial plant design and process modelling are based on the chemical composition of the samples, which is only a proxy for the mineralogical composition of the ores. A modelling method for mineral separation processes suited to complex ores should be particle-based, taking into consideration all quantifiable particle properties, and capable of estimating uncertainties. Moreover, to achieve a method generalizable to diverse mineral separation units (e.g., magnetic separation or flotation) with minimal human bias, strategies to independently weight the importance of different particle properties for the process(es) under investigation should be incorporated. This dissertation introduces a novel particle-based separation modelling method which fulfills these requirements. The core of the method consists of a least absolute shrinkage and selection operator-regularized (multinomial) logistic regression model trained with a balanced particle dataset. The required particle data are collected with scanning electron microscopy-based automated mineralogy systems. Ultimately, the method can quantify the recovery probability of individual particles, with minimal human input, considering the joint influence of particle shape, size, and modal and surface compositions, for any separation process. Three different case studies were modelled successfully using this new method, without the need for case-specific modifications: 1) the industrial recovery of pyrochlore from a carbonatite deposit with three froth flotation and one magnetic separation units, 2) the laboratory-scale magnetic separation of a complex skarn ore, and 3) the laboratory-scale separation of apatite from a sedimentary ore rich in carbonate minerals by flotation. Moreover, the generalization potential of the method was tested by predicting the process outcome of samples which had not been used in the model training phase, but came from the same geometallurgical domain of a specific ore deposit. In each of these cases, the method obtained high predictive accuracy. In addition to its predictive power, the new particle-based separation modelling method provides detailed insights into the influence of specific particle properties on processing behaviour. To name a couple, the influence of size on the recovery of different carbonate minerals by flotation in an industrial operation; and a comparison to traditional methodologies demonstrated the limitation of only considering particle liberation in process mineralogy studies -- the associated minerals should be evaluated, too. Finally, the potential application of the method to minimize the volume of test work required in metallurgical tests was showcased with a complex ore. The approach developed here provides a foundation for future developments, which can be used to optimize mineral separation processes based on particle properties. The opportunity exists to develop a similar approach to model the comminution of single particles and ultimately allow for the full prediction of the recovery potential of complex ores.:1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 State-of-the-art in particle-based separation models . . . . . . . . . . . 11 1.4 Moving forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Particle data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Mathematical tools required for the particle-based separation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.3 Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 The method and its application to industrial operations 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Assumptions and limitations . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Data structure and required pre-treatment . . . . . . . . . . . . 27 2.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.1 Artificial test cases . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.2 Real case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 Discussion and final considerations . . . . . . . . . . . . . . . . . . . . 39 3 The robustness of the method towards compositional variations of new feed samples 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Generalization potential of current Particle-based Separation Model (PSM) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Dry magnetic separation tests . . . . . . . . . . . . . . . . . . . 53 3.3.3 Sample characterization . . . . . . . . . . . . . . . . . . . . . . 53 3.3.4 Particle-based separation models . . . . . . . . . . . . . . . . . 54 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4 Flotation kinetics of individual particles 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.1 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Cumulative recovery probability . . . . . . . . . . . . . . . . . . 72 4.2.3 Particle-based kinetic flotation model . . . . . . . . . . . . . . . 74 4.3 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.1 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 Discussion and final thoughts . . . . . . . . . . . . . . . . . . . . . . . 80 5 Conclusions and outlook 85 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Bibliography 89