Rémy Bation
email: remy.bation@chu-lyon.fr

Marine Mondino

Florent Le Camus

Mohamed Saoud

Jerome Brunelin

Transcranial direct current stimulation in patients with obsessive compulsive disorder: A randomized controlled trial Authors

Background: Obsessive-compulsive disorder (OCD) is a severe mental disorder with poor response to the available treatments. Neuroimaging studies have identified dysfunctions within the orbito-fronto-striato-pallido-thalamic network in patients with OCD. Here, we assessed the efficacy and safety of transcranial direct current stimulation (tDCS) applied with the cathode over the orbitofrontal cortex (OFC) and the anode over the right cerebellum to decrease OCD symptoms in patients with treatment-resistant OCD.

Methods: In a randomized sham-controlled double-blind study, 21 patients with OCD were assigned to receive ten 20-min sessions (two sessions per day) of either active (2 mA) or sham tDCS. The clinical symptoms were measured using the Yale-Brown Obsessive and Compulsive Scale (YBOCS). Acute effects on the symptoms were measured from baseline to immediately after the 10 tDCS sessions. Long-lasting effects were measured 1 and 3 months after the 10th tDCS session.

Results: Compared with the sham tDCS, active tDCS significantly decreased OCD symptoms immediately after the 10th tDCS session (F(1,19)=5.26, p = 0.03). However, no significant differences were observed between the active and sham groups in terms of changes in YBOCS score or the number of responders one and 3 months after tDCS. Conclusion: Despite significant acute effects, tDCS with the cathode placed over the left OFC and the anode placed over the right cerebellum was not significantly effective in inducing a long-lasting reduction of symptoms in patients with treatment-resistant OCD.

Main Text

BACKGROUND

Obsessive-compulsive disorder (OCD) is a highly prevalent and devastating psychiatric condition. Despite advances in patient management, approximately 30-60% of patients do not improve or show a partial response to current medications whether given in combination with cognitive and behavioral therapies (CBTs) or are unable to tolerate medication side effects. [START_REF] Pallanti | Treatment-refractory obsessive-compulsive disorder: methodological issues, operational definitions and therapeutic lines[END_REF] Growing evidence suggests that the neurobiological substrates of OCD involve abnormal activity and connectivity within the orbito-fronto-striato-pallido-thalamic network with increased activity in the orbitofrontal cortex (OFC), the cingulate gyrus and the caudate [START_REF] Whiteside | A meta-analysis of functional neuroimaging in obsessive-compulsive disorder[END_REF][START_REF] Fettes | Cortico-Striatal-Thalamic Loop Circuits of the Orbitofrontal Cortex: Promising Therapeutic Targets in Psychiatric Illness[END_REF]. This hyperactivity is correlated with the severity of OCD and is diminished by effective pharmacological, psychotherapeutic and neurosurgical treatments [START_REF] Van Der Straten | Impact of treatment on resting cerebral blood flow and metabolism in obsessive compulsive disorder: a meta-analysis[END_REF]. Structural abnormalities have also been highlighted in this circuit in the form of decreased gray matter volume in the OFC and the cingulate, in contrast to the increased volume of the thalamus and the putamen [START_REF] Piras | Widespread structural brain changes in OCD: a systematic review of voxel-based morphometry studies[END_REF]. More recently, the role of the cerebellum in OCD physiopathology has been highlighted by studies showing increased cerebellar volume [START_REF] Pujol | Mapping structural brain alterations in obsessive-compulsive disorder[END_REF][START_REF] Tobe | Cerebellar morphology in Tourette syndrome and obsessive-compulsive disorder[END_REF][START_REF] Real | Stressful life events at onset of obsessive-compulsive disorder are associated with a distinct clinical pattern[END_REF][START_REF] Hu | Meta-analytic investigations of common and distinct grey matter alterations in youths and adults with obsessive-compulsive disorder[END_REF][START_REF] De Wit | Multicenter voxel-based morphometry mega-analysis of structural brain scans in obsessivecompulsive disorder[END_REF][START_REF] Eng | Meta-analytic investigations of structural grey matter, executive domain-related functional activations, and white matter diffusivity in obsessive compulsive disorder: an integrative review[END_REF] and increased connectivity [START_REF] Anticevic | Global restingstate functional magnetic resonance imaging analysis identifies frontal cortex, striatal, and cerebellar dysconnectivity in obsessive-compulsive disorder[END_REF][START_REF] Tian | Abnormal functional connectivity of brain network hubs associated with symptom severity in treatment-naive patients with obsessive-compulsive disorder: A resting-state functional MRI study[END_REF], including connections with the basal ganglia [START_REF] Vaghi | Specific Frontostriatal Circuits for Impaired Cognitive Flexibility and Goal-Directed Planning in Obsessive-Compulsive Disorder: Evidence From Resting-State Functional Connectivity[END_REF], in patients with OCD compared to those of healthy controls. Hypoactivity in the cerebellum has also been described in patients with OCD [START_REF] Nabeyama | Functional MRI study of brain activation alterations in patients with obsessive-compulsive disorder after symptom improvement[END_REF][START_REF] Wen | Pharmacotherapy response and regional cerebral blood flow characteristics in patients with obsessive-compulsive disorder[END_REF][START_REF] Milad | Deficits in conditioned fear extinction in obsessive-compulsive disorder and neurobiological changes in the fear circuit[END_REF][START_REF] Moon | Associations of neurofunctional, morphometric and metabolic abnormalities with clinical symptom severity and recognition deficit in obsessivecompulsive disorder[END_REF] and seems to be corrected by neurosurgical treatment [START_REF] Zuo | Metabolic imaging of bilateral anterior capsulotomy in refractory obsessive compulsive disorder: an FDG PET study[END_REF], cognitive and behavioral therapy [START_REF] Nabeyama | Functional MRI study of brain activation alterations in patients with obsessive-compulsive disorder after symptom improvement[END_REF][START_REF] Sanematsu | Predictors of treatment response to fluvoxamine in obsessive-compulsive disorder: an fMRI study[END_REF] or selective serotonin reuptake inhibitors (SSRIs) [START_REF] Sanematsu | Predictors of treatment response to fluvoxamine in obsessive-compulsive disorder: an fMRI study[END_REF][START_REF] Nakao | Brain activation of patients with obsessive-compulsive disorder during neuropsychological and symptom provocation tasks before and after symptom improvement: a functional magnetic resonance imaging study[END_REF]. Altogether, these studies suggest that the cerebellum might have a significant yet poorly appreciated role in neuropsychiatric diseases including OCD.

When used repeatedly, noninvasive brain stimulation techniques such as repetitive transcranial magnetic stimulation (rTMS) and direct current stimulation (tDCS) were described as powerful tools to modulate activity and connectivity within large widespread brain networks. The physiological effects of those techniques can outlast the stimulation period. In cases of treatment-resistant symptoms, these techniques have been proposed as therapeutic options when applied over abnormal brain regions in patients with several psychiatric conditions [START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of transcranial direct current stimulation (tDCS)[END_REF][START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS)[END_REF]. Hence, rTMS has been used in several clinical studies in patients with OCD [START_REF] Lusicic | Transcranial magnetic stimulation in the treatment of obsessive-compulsive disorder: current perspectives[END_REF]. Targeting the supplementary motor area (SMA), the OFC or the medial prefrontal cortex with "inhibitory" low-frequency rTMS holds promise for alleviating obsessive and compulsive symptoms [START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS)[END_REF][START_REF] Trevizol | Transcranial Magnetic Stimulation for Obsessive-Compulsive Disorder: An Updated Systematic Review and Meta-analysis[END_REF]. However, other studies failed to highlight the usefulness of rTMS in patients with OCD, leaving room to explore other approaches [START_REF] Lusicic | Transcranial magnetic stimulation in the treatment of obsessive-compulsive disorder: current perspectives[END_REF]. tDCS is a noninvasive technique in which a weak current is generated between an anode and a cathode placed over the scalp of the subject. Neurophysiological and animal studies described that an increase in cortical excitability and spontaneous neural activity occurred under the anode, whereas a decrease in cortical excitability and spontaneous neural activity occurred under the cathode [START_REF] Stagg | Physiology of Transcranial Direct Current Stimulation[END_REF]. tDCS enables the simultaneous stimulation of different areas and modulation of different brain areas involved in cortico-subcortical loops [START_REF] Fonteneau | Frontal Transcranial Direct Current Stimulation Induces Dopamine Release in the Ventral Striatum in Human[END_REF]. tDCS has been shown to have clinical effectiveness in the treatment of various psychiatric disorders [START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of transcranial direct current stimulation (tDCS)[END_REF]. It has been suggested that tDCS might be useful in decreasing the symptoms of patients with treatment-resistant OCD [START_REF] Brunelin | Transcranial Direct Current Stimulation for Obsessive-Compulsive Disorder: A Systematic Review[END_REF][START_REF] D'urso | Transcranial Direct Current Stimulation in Obsessive-Compulsive Disorder, Posttraumatic Stress Disorder, and Anxiety Disorders[END_REF], but the optimal target locations and stimulation parameters remain under debate.

We hypothesized that targeting the left hyperactive OFC (3) with cathodal tDCS (assumed to decrease cortical excitability (26)) coupled with anodal tDCS (assumed to increase cortical excitability [START_REF] Stagg | Physiology of Transcranial Direct Current Stimulation[END_REF]) over the right hypoactive cerebellum (15) may reduce obsessive and compulsive symptoms by modulating neuronal activity within the orbito-fronto-striatopallido-thalamic loop. We have previously reported the efficacy of this electrode montage in an open-label study that included 8 patients with treatment-resistant OCD [START_REF] Bation | Transcranial direct current stimulation in treatment-resistant obsessive-compulsive disorder: An open-label pilot study[END_REF]. Here, we developed a randomized sham-controlled study to assess the clinical efficacy and safety of OFC cathodal-tDCS coupled with cerebellum anodal-tDCS in patients with treatmentresistant OCD. We investigated the acute effects of tDCS by assessing the symptoms immediately after the tDCS regimen and the long-lasting effects by measuring symptoms one and three months after tDCS.

MATERIALS AND METHODS

Patients were randomly assigned in a 1:1 ratio to either the active or sham tDCS group. We used a block randomization with a block size of four. The study was approved by the local ethics committee (Comité de Protection des Personnes of Sud-Est 6, Lyon, France). All subjects gave their written informed consent after receiving a full description of the study.

This study was preliminarily registered on the clinicaltrials.gov database (clinicaltrials.gov identifier NCT 02407288).

Subjects

To be eligible, patients had to be 18-70 years old, have a primary diagnosis of OCD according to the DSM IV criteria, have clinically significant OCD symptoms (defined as a total Yale-Brown Obsessive and Compulsive Scale (YBOCS) (31) score > 16) and have received an adequate trial of SSRI (at least 12 weeks, of which 6 weeks had to be at the maximum tolerated dose). Patients were not included if they presented severe depressive symptoms defined by a Montgomery and Asberg Depression Rating Scale (MADRS) (32) score >30, had a history of psychotic disorders, drug abuse, or significant head injury, were acutely suicidal, or had implanted devices or metal in the brain. Pregnant women were not eligible to participate. Patients with prior exposure to tDCS were also excluded to reduce the risk of unblinding. Subjects were required to receive stable doses of a serotonin reuptake inhibitor The two daily sessions were separated by at least 3 hours. Each session of active tDCS consisted of delivering a direct current with an intensity set at 2 mA for 20 min. The electrodes were placed according to the international 10-20 electrode placement system. The cathode was placed on FP1 to target the left OFC, and the anode was placed 3 cm below the inion and 1 cm right of the midline to target the right cerebellum. To maintain blinding of the experimenters and participants regarding the tDCS condition, we used the commercial built-in sham procedure of the device (double-blind 'study mode'). The care provider entered a preprogrammed code that delivered either active or sham tDCS but was unaware of which condition the code applied. The list of codes was established by a researcher not involved in the tDCS delivery, data collection or analyses. In the sham, the same stimulation parameters as in the real condition were displayed, but after 40 s of real stimulation (2 mA), only brief current pulses of 110 μA over 15 ms occurred every 550 ms through the remainder of the 20min period. Participants, care providers, investigators, and outcome assessors were blinded to the treatment assignment. Safety was assessed after each tDCS session with a structured interview [START_REF] Brunoni | A systematic review on reporting and assessment of adverse effects associated with transcranial direct current stimulation[END_REF], and serious adverse events were systematically recorded by the investigator.

Clinical outcomes

All clinical outcomes were measured at baseline (before the first tDCS session), immediately after the 10 tDCS sessions (M0), and one month (M1) and three months after tDCS (M3).

Obsessive and compulsive symptoms were assessed using the Y-BOCS, the clinical global impression scale (CGI) [START_REF] William | Clinical Global Impressions[END_REF] and a self-reporting OCD visual analogue scale (OCD-VAS). To complete the OCD-VAS, patients were required to rate their current obsessive and compulsive symptoms on a 10-point scale (0 = Worst Ever to 10 = Best Ever). The clinical response was defined as a decrease ≥35% on the Y-BOCS or a score of 2 or less on the CGI-I (much or very much improved). Depressive symptoms and anxiety were assessed using the MADRS [START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF] and State-Trait Anxiety Inventory (STAI-YA) [START_REF] Spielberger | Manual for the State-trait anxiety inventory[END_REF], respectively. Level of functioning was assessed by the Global Assessment of Functioning (GAF) scale [START_REF] Pedersen | Revised manual for the Global Assessment of Functioning scale[END_REF].

Statistical analysis

Statistical analyses were performed using SPSS (version 22). Fisher's exact tests and twotailed Student's t-tests were applied to compare the demographics and clinical measures at baseline between the active and sham groups for categorical and continuous variables, respectively.

The primary analysis investigated acute changes in the YBOCS score. Acute changes in clinical scores between baseline and after the 10 tDCS sessions (at M0) were compared using repeated-measures analysis of variance (ANOVA) with treatment as the intergroup factor and time as the intrasubject factor. For significant changes, post hoc comparisons were performed using Fisher's least significant difference tests.

Secondary clinical analyses investigated maintenance effects on the YBOCS score, response rate and effects on other symptoms. Changes in the YBOCS scores over the 12-week followup period were compared using repeated-measures analysis of variance (ANOVA), including terms for group, time, and time by group interaction. Response rates at each time point (defined as an Y-BOCS score reduction of at least 35% between inclusion and month 1 post-tDCS) were compared using Fisher's exact test. The effects on other symptoms (MADRS, CGI, STAI, VAS) were analyzed in the same manner.

All tests were conducted with two-sided significance levels (α= 0.05).

RESULTS

Demographics and baseline clinical characteristics

Twenty-one patients were recruited and completed the trial. As shown in Table 1, the active and sham groups did not differ significantly in demographics or baseline clinical ratings.

INSERT TABLE 1 ABOUT HERE 2.2 Clinical outcomes

Clinical measures are presented in Table 2. The evolution of the total YBOCS score throughout the study period is represented in Figure 2.

Acute effect

The repeated-measures ANOVA revealed a significant interaction between time and group (F(1,19)=5.26, p = 0.033). After the 10 th session of tDCS, the YBOCS score increased by a mean of + 2.3% (standard deviation 5.9) in the sham group and decreased by -4.7% (7.9) in the active group. The post hoc comparisons showed a significant reduction in YBOCS scores in the active group after the 10 th session compared to those at baseline (p = 0.031). However, no differences were observed in the sham group between baseline and after the 10 th session, and no differences were observed between groups after the 10 th session.

Maintenance effect

The repeated-measures ANOVA for the 12-week period revealed a significant time effect (F(3,57) = 4.93, p = 0.01) but no significant group effect (F(3,57) = 0.64, p = 0.43) or interaction between time and group effect (F(3,57) = 0.66, p = 0.53).

No patients were classified as responders in either group at M0. The response rate at month 1 was not different between the 2 groups (10% in the active group vs 0% in the sham group; p= 0.47). The response rate at month 3 was not different between the 2 groups (20% in the active group vs 10% in the sham group, p = 0.58). Two patients in the active group were classified as responders at the end of the trial. These two patients had the lowest level of resistance (2 and 3 according to the stage of resistance defined by Pallanti and Quercioli (1): 2 and 3 SSRIs and a cognitive and behavioral therapy method) in the active group (mean level of resistance: 6.2) and were both young women with contamination OCD.

Effect on other symptoms

The repeated-measures ANOVA on the Global Assessment Functioning scores revealed a significant group by time interaction (p=0.028) (Table 2), but the post hoc tests showed no significant differences between the groups at any time.

The linear regression model testing if changes in the total Y-BOCS scores between baseline and month 3 post-tDCS were associated with resistance level, total YBOCS score at baseline, depression score at baseline, anxiety score at baseline, sex, and thematic content revealed that only the level of resistance was associated with an improvement in YBOCS score at the end of the trial (F(1,9) = 7.695, R 2 = 0.490 p=0.024) in the active group.

Regarding the safety of tDCS in patients with OCD, the stimulation sessions were well tolerated by the patients, and no serious adverse events were observed.

INSERT FIGURE 2 ABOUT HERE INSERT TABLE 2 ABOUT HERE 3 DISCUSSION

The present study is the first parallel randomized sham-controlled study to explore the clinical efficacy and safety of fronto-cerebellar tDCS in patients with SSRI-resistant OCD. The results indicated that active tDCS targeting the left OFC and the right cerebellum induced a significant acute reduction of OCD symptoms immediately after the tDCS regimen compared to sham stimulation. However, active tDCS was not superior to sham stimulation in alleviating OCD symptoms over a 12-week period. Among the different clinical measures evaluated (total YBOCS scores, subscores, questionnaire results, clinician global inventory score), no significant differences between the active and sham groups were revealed.

The lack of significant long-lasting effects of active tDCS observed in this study could be partially explained by the high level of resistance of the included subjects. The majority of subjects included in this study had severe (YBOCS score at inclusion of 29 and 29.5 for the active and sham groups, respectively), chronic (an illness duration of more than 20 years) and resistant symptoms (mean resistance level above 5, which corresponds to at least 3 selective serotonin reuptake inhibitors, including clomipramine and a cognitive and behavioral therapy method). The patients were on a stable medication dosage for at least six weeks before inclusion into the study and during the three-month follow-up. Assessments were performed in accordance with the European Medicines Agency guidelines with the OCD auto-and hetero-evaluations (OCD-VAS and YBOCS), a functional assessment (CGI) and a threemonth follow-up [START_REF] Medecines | Clinical investigation of medicinal products for the treatment of obsessive compulsive disorder[END_REF]. The population was carefully selected to be clinically stable without major comorbidities, including depression (MADRS < 30). A linear regression confirmed that the level of resistance was related to improvements in obsessive and compulsive symptoms at the end of the trial. In line with this hypothesis, the two responders had the lowest resistance level within the active group. Thus, these negative results cannot be easily generalized to SSRI-resistant OCD patients because our population showed a high level of resistance (above 5 according to the Pallanti and Quercioli criteria (1)).

Second, the lack of significant long-lasting effects observed in the current study can also be partially explained by the tDCS parameters. Indeed, the optimal tDCS parameters in terms of both dosage and electrode placement are not yet well-established for the treatment of OCD [START_REF] Senço | Transcranial direct current stimulation in obsessive-compulsive disorder: emerging clinical evidence and considerations for optimal montage of electrodes[END_REF] and could have been suboptimal in our study. We chose to use a treatment pattern that has induced beneficial outcomes in patients with schizophrenia [START_REF] Brunelin | Examining transcranial direct-current stimulation (tDCS) as a treatment for hallucinations in schizophrenia[END_REF]: two 20-min daily sessions delivered in one week; however, recent data showed that patients with other psychiatric diseases (e.g., depression and schizophrenia) favored longer treatment (>4 weeks in depression), higher intensity (3 mA versus 2 mA) and longer sessions (30 min versus 20 min). One can hypothesize that a higher tDCS dosage than the dosage in this study could be suitable to achieve remission in patients with OCD [START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of transcranial direct current stimulation (tDCS)[END_REF]. The tDCS dose, along with head anatomy, determines the resulting current flow (intensity and pattern) in the brain and in turn affects the resulting neurophysiological and behavioral changes [START_REF] Truong | Physics of Transcranial Direct Current Stimulation Devices and Their History[END_REF]. An electric field distribution simulation made using SimNIBS 2.0.1 [START_REF] Thielscher | Field modeling for transcranial magnetic stimulation: A useful tool to understand the physiological effects of TMS?[END_REF] of the electrode montage used in the current study showed a high electric density in the right cerebellum, but the electric field in the prefrontal cortex was more pronounced in the inferior frontal gyrus than in the orbitofrontal gyrus (Figure 3). The insufficient electric field density in the OFC area might explain the lack of long-lasting clinical effects observed in the current study.

INSERT FIGURE 3 ABOUT HERE

Our montage is the first cerebellar tDCS montage used in OCD patients. The cerebellar involvement in OCD pathophysiology has gained increasing attention in recent years with large-scale studies and meta-analyses showing bilateral enlargement and functional and structural hyperconnectivity in the cerebellum [START_REF] Lázaro | Microstructural brain abnormalities and symptom dimensions in child and adolescent patients with obsessivecompulsive disorder: a diffusion tensor imaging study[END_REF][START_REF] Hartmann | Increased fractional anisotropy in cerebellum in obsessive-compulsive disorder[END_REF]. However, specifying the nature of cerebellar dysfunction remains a daunting task. According to the embodied cognition theory, cognition and emotion are grounded in the sensorimotor system and share similar modes of processing. These connections might be extremely relevant in the cerebellar setting where sensorimotor, cognition and emotion share the same type of processing (Universal Cerebellar transform theory [START_REF] Guell | Embodied cognition and the cerebellum: Perspectives from the Dysmetria of Thought and the Universal Cerebellar Transform theories[END_REF]). Therefore, the cerebellum and its connection with the basal ganglia and cortical area may implement habitual, nonflexible patterns of movement (rituals), cognitions and emotions (obsessions) [START_REF] Watson | Conflicted between Goal-Directed and Habitual Control, an fMRI Investigation[END_REF]. The cerebellum and its connection with basal ganglia loops are involved in setting habits [START_REF] Watson | Conflicted between Goal-Directed and Habitual Control, an fMRI Investigation[END_REF] and could, in this way, contribute to the imbalance between goal-directed actions and habits in OCD [START_REF] Gillan | Habits and goals in OCD. In: Obsessive-compulsive disorder: Phenomenology, pathophysiology, and treatment[END_REF]. Volume alteration has not been reported in children with OCD [START_REF] Hu | Meta-analytic investigations of common and distinct grey matter alterations in youths and adults with obsessive-compulsive disorder[END_REF], suggesting that cerebellum enlargement could be an adaptive process of OCD tendencies to over-rely on habitual rigid automatic stimulus-driven behaviors and cognitions.

In summary, tDCS with the cathode placed over the left OFC combined with the anode placed over the right cerebellum is a safe approach in patients with OCD, but despite a statistically significant acute effect, we failed to confirm the promising results observed in a previous open-label trial [START_REF] Bation | Transcranial direct current stimulation in treatment-resistant obsessive-compulsive disorder: An open-label pilot study[END_REF]. Many positive results obtained in open label study fail to be replicated in RCT. Different phenomenon may increase the risk of false positive in open label study: publication bias, regression towards the average, a higher level of expectancy for response compare to the level of expectancy in RCT. One limitation of the current study is the sample size. Indeed, although comparable with other studies in the field, it could be considered as relatively small, especially in the context of heterogeneity in OCD. Owing to the highly prevalent resistance of OCD patients in this study, further controlled studies should be considered before any conclusions are made regarding the usefulness of tDCS in patients with OCD. Future studies should pay particular attention to factors that can improve the clinical efficacy of tDCS. First, inducing activity during stimulation of the targeted neuronal networks is of prime importance to achieving neurobiological and clinical effects because tDCS mostly works by enhancing neuroplasticity [START_REF] Huang | Plasticity induced by non-invasive transcranial brain stimulation: A position paper[END_REF] and learning [START_REF] Simonsmeier | Electrical brain stimulation (tES) improves learning more than performance: A meta-analysis[END_REF]. Combining noninvasive brain stimulation with symptom provocation [START_REF] Carmi | Clinical and electrophysiological outcomes of deep TMS over the medial prefrontal and anterior cingulate cortices in OCD patients[END_REF][START_REF] Fda | Press Announcements -FDA permits marketing of transcranial magnetic stimulation for treatment of obsessive compulsive disorder[END_REF] or cognitive behavioral therapy techniques could potentially improve outcomes [START_REF] Sathappan | The dynamic Duo: Combining noninvasive brain stimulation with cognitive interventions[END_REF]. Second, the amount of electrical stimulation that reaches the targeted area is a critical factor in inducing clinical effects. Computational models to estimate cortical electric fields [START_REF] Edwards | Physiological and modeling evidence for focal transcranial electrical brain stimulation in humans: a basis for high-definition tDCS[END_REF] and high-definition tDCS [START_REF] Turski | Extended Multiple-Field High-Definition transcranial direct current stimulation (HD-tDCS) is well tolerated and safe in healthy adults[END_REF] are tools that may help us design more focused tDCS montages. Third, the tDCS dosage could be insufficient in our study. Using a higher number of sessions over a longer period of time may be needed to obtain clinically significant responses. Lastly, optimal electrode positioning in OCD remains a daunting issue. Neuroimaging studies, can only infer correlation and not causation. Noninvasive brain stimulation clinical studies have yielded mixed findings, as distinct brain areas have been stimulated, (e.g., the left or right dorsolateral prefrontal cortex, the OFC, the supplementary motor areas) with no distinct most effective intervention [START_REF] Lusicic | Transcranial magnetic stimulation in the treatment of obsessive-compulsive disorder: current perspectives[END_REF]. Two categories of tDCS montages have been used: montages with electric field focused in different prefrontal cortical regions and montages with more diffused electric field in larger and distributed cortical areas across different brain regions. However, for both categories of montages improvement of symptoms has been reported [START_REF] Da Silva R De | Transcranial direct current stimulation in obsessive-compulsive disorder: an update in electric field modeling and investigations for optimal electrode montage[END_REF]. The incomplete understanding of the neurocircuits involved in OCD and its different subtypes (55) associated with the variability of individual effects of tDCS may explain modest and heterogenous clinical results in resistant OCD.

Conflicts of interest

None to disclose

Funding

The study was supported by the "Conseil Scientifique de la Recherche" from CH le Vinatier.

(

  Figure 1.

Figure 1 .

 1 Figure 1. CONSORT flow chart diagram of the study.

Figure. 2 .

 2 Figure. 2. Evolution of the Y-BOCS scores in patients with OCD after receiving either active

Figure 3 .

 3 Figure 3. Estimation results of the electric field distribution of the tDCS montage with two 7 x

♦

  Analysed (n=10) ♦ Excluded from analysis (n=0) Lost to follow-up (n=0 ) Discontinued intervention (n=0 ) Allocated to intervention (n=10) ♦ Received allocated intervention (n=10) ♦ Did not receive allocated intervention (n=0 ) Lost to follow-up (n=0 ) Discontinued intervention (give reasons) (n=0 ) Allocated to intervention (n=11) ♦ Received allocated intervention (n= 11) Did not receive allocated intervention (n=0) Analysed (n=11 ) ♦ Excluded from analysis (n=0) )

Table 1 :

 1 Demographic and clinical baseline characteristics of the patients with OCD included in the study. No differences were observed between patients in the active and sham groups.The results are given as the mean (SD: standard deviation)

	Active tDCS	Sham tDCS
	Mean	SD	Mean	SD	p

Table . 2

 . : Changes in clinical outcomes after patients received 10 sessions of tDCS (ANOVA with repeated measures) *: p≤0.05 .98 13.60 ± 2.32 13.30 ± 3.23 15.36 ± 3.59 15.45 ± 3.27 14.55 ± 3.67 14.64 ± 2.46 F(3.57)=0.697; p=0.59 F(3.57)=3.288; p=0.03* YBOCS compulsion 14.10 ± 2.08 13.30 ± 1.42 12.10 ± 1.85 12.50 ± 2.84 14.09 ± 3.94 14.55 ± 3.67 13.09 ± 3.67 13.64 ± 3.75 .00 ± 1.05 5.20 ± 1.40 6.00 ± 0.89 5.45 ± 1.44 5.91 ± 0.83 5.82 ± 0.87 F(3.57)=0.112; p=0.71 F(3.57)=2.308; p=0.11 MADRS 13.30 ± 6.95 11.00 ± 7.53 11.30 ± 6.72 13.50 ± 7.28 18.36 ± 5.30 19.00 ± 6.62 15.73 ± 6.92 15.73 ± 6.81 F(3.57)=1.059; p=0.37 F(3.57)=0.693; p= 0.56 STAI-YA 47.63 ± 18.07 45.90 ± 9.96 51.22 ± 14.29 49.30 ± 14.67 52.27 ± 8.47 46.00 ± 13.74 53.18 ± 11.32 53.45 ± 9.49 F(3.45)=0.127; p=0.94 F(3.45)=1.459; p=0.24 GAF 42.30 ± 13.33 43.20 ± 13.70 44.40 ± 13.53 47.20 ± 14.61 41.71 ± 14.10 40.27 ± 12.05 46.00 ± 12.32 40.64 ± 12.26 F(3,57) =3.248. p=0.028* F(3,57) =2.862. p=0.045* M0: after 10 sessions of tDCS; M1: one month after tDCS, and M3: 3 months after tDCS

				Active tDCS (N=10)			Sham tDCS (N=11)
		BASELINE		M0	M1	M3	Baseline	M0	M1	M3	ANOVA (time*group)	ANOVA (time)
	YBOCS total	29.00 ± 4.55	27.50 ± 3.95 25.70 ± 3.56 25.90 ± 5.84 29.45 ± 6.65 30.00 ± 6.29 27.45 ± 6.52 28.27 ± 5.82	F(3.57)=0.663; p=0.53	F(3.57)=4.927; p= 0.01*
	responders (%)		0.00	10.00	20.00			0.00	0.00	9.09
	YBOCS
	obsession	15.00 ± 2.75	14.00 ± 2F(3.57)=0.64; p=0.53	F(3.57)=3.889; p=0.03*
	VAS	6.92 ± 1.50	6.29	± 1.38 6.68 ± 2.04 6.40 ± 2.27 6.56 ± 2.03 6.59 ± 1.17 5.98 ± 1.76 6.29 ± 2.00	F(3.54)=0.779; p=0.51	F(3.54)=0.486; p= 0.69
	CGI-S	6.10 ± 0.99	6.00	± 1.05 6Assessed for eligibility (n=35)
							Excluded (n=14)
							•	Not meeting inclusion criteria (n=1)
							•	Declined to participate (n=13
								o only interested in open-label
								research studies (n= 8)
								o inability to take time off of
								work (n=4)
								o inability to travel to the
							•	clinical center (n=1) Other reasons (n=0)

Acknowledgments

The authors thank Caroline Damasceno, Frederic Haesebaert, and Philippe Vignaud for their help with the project.


Y Foucaud
email: yann.foucaud@univ-lorraine.fr

M Badawi

L Filippov

I Filippova

S Lebègue

A review of atomistic simulation methods for surface physicalchemistry phenomena applied to froth flotation

Keywords: Adsorption, water, mineral, Density Functional Theory, ab initio, molecular dynamics, collectors, depressants

The froth flotation method, which is used for the processing of most non-ferrous ores, involves the adsorption of both organic and inorganic reagents at the mineral/water interface. Understanding the adsorption mechanisms of flotation reagents is a key step to enhance the flotation. New depressants and collectors formulations, more efficient, selective, and environmental friendly can be suggested. At the moment, few experimental methods can show surface molecular mechanisms with accuracy and confidence. Therefore, atomistic simulations allow to gain understanding in the mechanisms involved in the reagents adsorption. Nowadays, atomistic simulations can be applied to describe the solid/liquid interface and the adsorption mechanisms. Depending on the method, the interactions between atoms are described on the basis of density functional theory (DFT) or force-fields. They give access to various levels of accuracy, parameters consideration, sampling times, system sizes, and reactivity description, depending on the method used. In particular, ab initio molecular dynamics allows to investigate adsorption processes at liquid-solid interfaces at finite temperature with accuracy.

Introduction

A. The froth flotation process

The world demand in mineral commodities, including metals and industrial minerals, has been exponentially increasing over the past decades. Although physical separations are widely used to concentrate the target metal-bearing mineral(s), most of non-ferrous ores are, today, processed by the froth flotation method [START_REF] Leja | Surface Chemistry of Froth Flotation[END_REF][START_REF] Bulatovic | Handbook of flotation reagents: chemistry, theory and practice[END_REF]Fuerstenau et al., 2007;[START_REF] Nguyen | Froth Flotation[END_REF]. This method is based on the differences of surface properties between the target mineral(s), i.e. the metalbearing mineral(s), and the gangue minerals, which have to be rejected. Performed on size fractions ranging from 10 µm to 300 µm [START_REF] Leja | Surface Chemistry of Froth Flotation[END_REF]Fuerstenau et al., 2007;[START_REF] Ralston | Reducing uncertainty in mineral flotation-flotation rate constant prediction for particles in an operating plant ore[END_REF][START_REF] Fornasiero | Innovations in the flotation of fine and coarse particles[END_REF], the froth flotation uses the specific adsorption of amphiphilic molecules called collectors onto the target mineral(s) surfaces to make them hydrophobic. The injection of air bubbles in the flotation cell under strong stirring conditions allows to recover the hydrophobic mineral(s) in the froth, at the upper part of the cell. The flotation collectors are commonly composed of a polar function, which adsorbs onto the solid, and a non-polar aliphatic chain that make the particle hydrophobic. The polar function can be either a cationic group such as amine function, which is assumed to physisorb onto the mineral surfaces [START_REF] Fuerstenau | CHAPTER K PRINCIPLES OF MINERAL FLOTATION[END_REF][START_REF] Leja | Surface Chemistry of Froth Flotation[END_REF][START_REF] Bulatovic | Handbook of flotation reagents: chemistry, theory and practice[END_REF], or an anionic group, which establishes covalent bond(s) with surface cations [START_REF] Fuerstenau | CHAPTER K PRINCIPLES OF MINERAL FLOTATION[END_REF][START_REF] Fuerstenau | Anionic flotation of oxides and silicates[END_REF][START_REF] Leja | Surface Chemistry of Froth Flotation[END_REF][START_REF] Bulatovic | Handbook of flotation reagents: chemistry, theory and practice[END_REF]Fuerstenau et al., 2007;Foucaud et al., 2018b). Depending on the target mineral(s), the anionic polar function can be carboxylic acids/alkali carboxylates, sulphate, sulfonate, xanthate, dithiophosphate... The collector adsorption depends strongly on the pH of the solution since the physisorption of cationic collectors is related to the zeta-potential while the surface hydroxylation can prevent the chemisorption of anionic collectors. Moreover, the collector adsorption can be modulated by adding a large range of molecules (organic or minerals) called depressants that keep the surface hydrophilic by preventing the collector adsorption and establishing hydrogen bonds with bulk water molecules. Finally, the flotation performances have been significantly increased over the past few years by mixing collectors with different aliphatic chains and/or with different polar groups as well as different depressants [START_REF] Filippov | Selective flotation of silicates and Ca-bearing minerals: The role of non-ionic reagent on cationic flotation[END_REF][START_REF] Filippova | Synergetic effect of a mixture of anionic and nonionic reagents: Ca mineral contrast separation by flotation at neutral pH[END_REF][START_REF] Filippov | Rare earths (La, Ce, Nd) and rare metals (Sn, Nb, W) as by-products of kaolin production -Part 3: Processing of fines using gravity and flotation[END_REF][START_REF] Filippov | New reagent formulations for selective flotation of scheelite from a skarn ore with complex calcium minerals gangue[END_REF][START_REF] Foucaud | Investigation of the depressants involved in the selective flotation of scheelite from apatite, fluorite, and calcium silicates: Focus on the sodium silicate/sodium carbonate system[END_REF].

B. Importance of molecular modelling for flotation

Liberations sizes in most ores are nowadays low enough to make unable the use of classical physical separation methods, making the froth flotation mandatory to concentrate the target mineral(s). Also, the average ore grades are decreasing over time [START_REF] Mudd | Global trends in gold mining: Towards quantifying environmental and resource sustainability[END_REF][START_REF] Calvo | Thermodynamic model of ionic and nonionic surfactants adsorptionabstraction on heterogeneous surfaces[END_REF], which can be mostly attributed to the depletion of the high grade deposits, although some authors interpret this decline as innovation and improvements in extractive technologies [START_REF] Drielsma | Abiotic Raw-Materials in Life Cycle Impact Assessments: An Emerging Consensus across Disciplines[END_REF]. Along with the ore grades decline, ores displaying high separation contrast between the constitutive minerals will also decline. Since the froth flotation is mandatory for most ores, the gangue complexness as well as the low grades will be overcome by increasing the flotation performances, i.e. the selectivity between minerals and the flotation recovery. Hence, understanding the molecular mechanisms involved in the adsorption of flotation reagents (depressants, activators, and collectors) is of paramount interest. The determination of adsorption energies and structural configurations allows to adapt the reagents used during the flotation process to the ore mineralogical association. Moreover, it brings clarity in the comprehension of the synergistic effects that are commonly exhibited when reagents are used in combination. New reagents formulations can be developed based on the adsorption energies calculated from molecular simulations, which allow to improve the flotation selectivity or recovery. Because of the presence of a liquid phase, the molecular mechanisms are difficult to investigate with experimental methods, mainly when similar reagents are used in mixtures, which makes the molecular modelling a powerful tool.

Over the past few years, molecular modelling has gathered interest to understand the key mechanisms that are the basis of the froth flotation process. This increase can be assessed by expressing the number of original research articles that have included the words "flotation" and "molecular simulations" over the fifteen last years (Fig. 1a). The energy of systems describing the adsorption processes can be calculated using highly accurate methods based on quantum mechanical descriptions such as Hartree-Fock, post-Hartree-Fock, and density functional theory (DFT) [START_REF] Sholl | Investigating the Quartz (101̅ 0)/Water Interface using Classical and Ab Initio Molecular Dynamics[END_REF][START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF]. Besides, energies and structures can be determined using classical potentials-based simulations that require accurate force fields and are not able to fully describe quantum mechanisms such as chemical bond cleavage or formation [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF]. Most of the above-mentioned articles used DFT (Fig. 1b), which exhibits the best compromise between accuracy and computational demand among the current molecular modelling methods [START_REF] Sholl | Investigating the Quartz (101̅ 0)/Water Interface using Classical and Ab Initio Molecular Dynamics[END_REF]. In the present review, we report the different molecular simulation methods existing nowadays in theoretical physics and chemistry and that have been successfully used in the flotation field. Each method is described with its advantages and limitations and some model systems are given as examples. Finally, a global method for modelling the interactions between flotation reagents and mineral surfaces is suggested.

Quantum mechanical simulations

In quantum simulations, it is assumed that all the bonds and interactions between atoms are related to the electrons, which can be described using the Schrödinger equation, considered as time-independent to simplify the problem:

, … , , … , , , … , = , … , , … , , , … , (1)
Where represents the total Hamiltonian operator of the system, the wave function that is a function of the spatial coordinates of nuclei (R) and electrons (r) and the spin of electrons (σ) that will be ignored for the rest of the discussion, and E the total energy of the system. In equation ( 1), the Hamiltonian operator can be expressed as the sum of total kinetic energy operator and the Colombian interactions. The kinetic energy operator takes into account the electrons kinetic energy operator, , and the nuclear kinetic energy operator, , while the Colombian interactions include the electronelectron repulsion ( ), the electron-nucleus attraction ( ), and the nucleus-nucleus repulsion ( ) operators:

= + + + + (2)
Hence, the time-independent Schrödinger equation depends on 4N+3M variables, where N is the number of electrons and M the number of nuclei, which makes nearly impossible a numerical resolution. To reduce the number of variables involved in the Schrödinger equation, the Born-Oppenheimer approximation is commonly used, in which the nuclei are considered immobile since their mass is significantly higher than the electrons (around 1800 times). Therefore, the nuclear kinetic energy operator can be neglected while the nucleus-nucleus Colombian interaction is constant. It induces that the total energy of the system corresponds to the sum of the total electronic energy of the system and a constant term representing the nucleus-nucleus interaction term.

In the Born-Oppenheimer approximation, the electron-electron repulsion term, still does not have a simple analytical solution for systems with more than 2 electrons. The Hartree-Fock method [START_REF] Slater | Note on Hartree's Method[END_REF][START_REF] Hartree | Self-consistent field, with exchange, for beryllium[END_REF] was developed to solve this problem, in which the wave function is expressed as a Slater determinant and the Schrödinger equation is solved self-consistently to minimize the total energy of the system [START_REF] Nesbet | Variational Principles and Methods in Theoretical Physics and Chemistry[END_REF][START_REF] Epstein | The variation method in quantum chemistry[END_REF]. Although the exchange is correctly described by this method, the correlation is not taken into account, which induces an energy overestimation. Hence, post-Hartree-Fock methods [START_REF] Cramer | Essentials of computational chemistry: theories and models[END_REF][START_REF] Jensen | Introduction to computational chemistry[END_REF] were developed to take into account correlation terms, resulting in accurate energy determinations but high computational costs that make this method viable only for small systems.

Besides, Hohenberg and Kohn demonstrated that the total energy E0 of a system in its fundamental state can be expressed as a functional that depends only on the electronic density [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF]. In this formalism, the total energy of the system (E) is the sum of the kinetic energies of the electrons T[ρ], the repulsive Colombian electron-electron interactions J[ρ], the electron-nucleus interaction potential Vne( ), and a term corresponding to the electron-electron exchange-correlation effects Encl[ρ]:

= + + + (3)
To propose a practical mean to solve this equation, Kohn and Sham suggested a method [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] based on an imaginary system in which the electrons do not interact between themselves and move in a potential Vs that is chosen so that the calculated electronic density of the imaginary system is equal to the real electronic density.

The unknown terms are gathered in a quantity called the exchange-correlation energy, Exc[ρ], which cannot be determined analytically in the Kohn-Sham equations. A number of exchange-correlation functionals have been developed to calculate the exchange-correlation energy, for instance in the local density approximation (LDA) or in the generalized gradient approximation (GGA). LDA approximates the system by a uniform homogenous electrons gas [START_REF] Perdew | Self-interaction correction to density-functional approximations for many-electron systems[END_REF][START_REF] Hafner | Ab-initio simulations of materials using VASP: Density-functional theory and beyond[END_REF] while GGA takes into account the gradient density and, thus, is more accurate for systems exhibiting significant density variations [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF][START_REF] Johnson | The performance of a family of density functional methods[END_REF][START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF]. Besides, GGA is known to not describe precisely dispersion interactions, which represent a part of van der Waals forces [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF]. Consequently, methods have been intensively developed to correct the dispersion interactions by adding a term in the exchange-correlation functional with various degrees of accuracy [see for instance [START_REF] Grimme | Semiempirical GGA-type density functional constructed with a long-range dispersion correction[END_REF][START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF][START_REF] Bučko | Improved Description of the Structure of Molecular and Layered Crystals: Ab Initio DFT Calculations with van der Waals Corrections[END_REF]Bučko et al., , 2013aBučko et al., , 2013b[START_REF] Bučko | Extending the applicability of the Tkatchenko-Scheffler dispersion correction via iterative Hirshfeld partitioning[END_REF][START_REF] Bučko | Many-body dispersion corrections for periodic systems: an efficient reciprocal space implementation[END_REF]]. Dispersion interactions should be included in the DFT calculations since the adsorption of flotation reagents, mostly collectors, is partly led by hydrophobic interactions between the aliphatic chains (Cases and Villieras, 1992).

Overall, DFT simplifies a 3N-dimensional problem (where N is the number of electrons) to a 3-dimensional problem with a high accuracy and a relatively low computational cost. Hence, it enables the electronic structure calculations of crystals and molecules in liquid or gas phases for relatively large systems. More than 15 000 papers including DFT are published each year in various fields including condensed matter physics [START_REF] Csonka | Assessing the performance of recent density functionals for bulk solids[END_REF][START_REF] Lebègue | Electronic structure of two-dimensional crystals from ab initio theory[END_REF][START_REF] Bučko | Improved Description of the Structure of Molecular and Layered Crystals: Ab Initio DFT Calculations with van der Waals Corrections[END_REF][START_REF] Srour | Comparative study of structural and electronic properties of GaSe and InSe polytypes[END_REF], interface phenomena [START_REF] Bučko | Dissociative iodomethane adsorption on Ag-MOR and the formation of AgI clusters: an ab initio molecular dynamics study[END_REF][START_REF] Chebbi | Evaluation of volatile iodine trapping in presence of contaminants: A periodic DFT study on cation exchanged-faujasite[END_REF][START_REF] Chibani | A DFT study of RuO 4 interactions with porous materials: metal-organic frameworks (MOFs) and zeolites[END_REF][START_REF] Jabraoui | Adsorption of volatile organic and iodine compounds over silver-exchanged mordenites: A comparative periodic DFT study for several silver loadings[END_REF], catalysis [START_REF] Badawi | DFT study of furan adsorption over stable molybdenum sulfide catalyst under HDO conditions[END_REF]M. Badawi et al., 2011a;[START_REF] Badawi | Guaiacol derivatives and inhibiting species adsorption over MoS2 and CoMoS catalysts under HDO conditions: A DFT study[END_REF][START_REF] Badawi | Hydrodeoxygenation of Phenolic Compounds by Sulfided (Co)Mo/Al 2 O 3 Catalysts, a Combined Experimental and Theoretical Study[END_REF])… This method displays now a very good reproducibility that has been assessed recently on 71 elemental crystals [START_REF] Lejaeghere | Reproducibility in density functional theory calculations of solids[END_REF] as well as very good agreement with high-precision experiments [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF][START_REF] Losch | Proton Mobility, Intrinsic Acid Strength, and Acid Site Location in Zeolites Revealed by Varying Temperature Infrared Spectroscopy and Density Functional Theory Studies[END_REF]. The global development of high-performance computing (HPC) allows to perform DFT calculations on systems with up to 1500-2000 atoms in around 50 Å 3 cells [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF].

Molecular dynamics simulations techniques

DFT calculations are used to determine the structure and energy of a system in its ground-state and, hence, do not describe atoms movements and the thermal motion. For that, molecular dynamics (MD)

simulations can be used: they produce dynamical trajectories for a system composed of N particles by integrating Newton's equations of motion:

! " ! # " = $ ! = - & & ! ' , " , … , (4)
Where fi represents the external forces acting on the particle i of mass mi and position ri, which corresponds to the negative of the gradient of the potential energy of the system. The external forces are calculated at each time step, which should be small enough to describe correctly the fastest intramolecular vibrational motions, so usually around 1 fs. The forces can be calculated either by means of DFT, which is called the ab initio molecular dynamics (AIMD) simulations, or by means of force fields (FF), which is called classical molecular dynamics (CMD) simulations. Both of these simulations are performed at constant number of particles (N), volume of the simulation cell (V), and temperature (NVT), pressure (NVP), or energy (NVE). For that, various thermostats or barostats, respectively, are available [START_REF] Hünenberger | Thermostat Algorithms for Molecular Dynamics Simulations[END_REF][START_REF] González | Force fields and molecular dynamics simulations[END_REF] and enable the comparison between the simulation results and experiments. Notably, the Nosé-Hoover thermostat is among the most widely used thermostats since it considers the heat bath as a part of the system, with an additional artificial variable and an associated mass [START_REF] Nosé | A molecular dynamics method for simulations in the canonical ensemble[END_REF][START_REF] Nosé | A unified formulation of the constant temperature molecular dynamics methods[END_REF][START_REF] Hoover | Canonical dynamics: Equilibrium phase-space distributions[END_REF].

Both CMD and AIMD will be discussed with upsides and downsides given for each method along

with successful examples of use.

A. Classical molecular dynamics (CMD) simulations CMD are based on force fields, which enable the calculation of the external forces acting on each particle for the integration of Newton's equations of motion. Force fields represent an expression of the energy of a system as a function of the coordinates of its particles [START_REF] González | Force fields and molecular dynamics simulations[END_REF]. They include several terms that can be divided into three main groups: intramolecular, intermolecular, and special terms. The first ones mainly describe the bonds, angles, torsions, etc., the second ones refer to Coulombic and van der Waals interactions between atoms, while the special terms may include crossterms or reactivity terms. Many different force fields have been developed over the past decades, among which some were designed to simulate the liquid/solid interfaces such as OPLS [START_REF] Jorgensen | Development and Testing of the OPLS All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids[END_REF] or COMPASS [START_REF] Sun | COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase ApplicationsOverview with Details on Alkane and Benzene Compounds[END_REF]. Most force fields were developed based on ab initio calculations, which were used to determine the parameters acting on each term. The force field choice should be adapted to the modelled system since each force field exhibits particular strengths and weaknesses, depending on the parametrization. However, most force fields do not include any term for the reactivity, i.e. bond creation or destruction, which cannot be described accurately by simple force models since they result from electronic processes [START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF]. However, CMD simulations 2001). Overall, CMD can be performed on systems containing millions of atoms [START_REF] Freddolino | Molecular Dynamics Simulations of the Complete Satellite Tobacco Mosaic Virus[END_REF]Kadau et al., 2006;[START_REF] Klepeis | Long-timescale molecular dynamics simulations of protein structure and function[END_REF][START_REF] Schulz | Scaling of Multimillion-Atom Biological Molecular Dynamics Simulation on a Petascale Supercomputer[END_REF][START_REF] González | Force fields and molecular dynamics simulations[END_REF][START_REF] Evans | Molecular Design of Amorphous Porous Organic Cages for Enhanced Gas Storage[END_REF][START_REF] Evans | Computational Chemistry Methods for Nanoporous Materials[END_REF] with total simulations times reaching the order of microseconds, which enables the investigation of macroscopic properties such as the diffusion of molecules in liquid phase.

B. Ab initio molecular dynamics (AIMD) simulations

In Eq. 4, the external forces acting the particles can be determined by a DFT calculation since the total energy of the system can be easily related to the total forces. This allows to perform a MD simulation at finite-temperature with a DFT calculation for each time step. However, the quantum chemical calculation realized at every time step limits significantly the system size [START_REF] Kühne | Second generation Car-Parrinello molecular dynamics: Second-generation Car-Parrinello molecular dynamics[END_REF]. Nowadays, AIMD simulations are one of the most powerful methods for energy and structure determination as they associate the high accuracy of DFT and the temperature/pressure control of molecular dynamics simulations. Since the electrons are included, the chemical bonds can break and form, which enables the study of chemical-based molecular processes and reactivity. Today, with the development of HPC, AIMD simulations can reach total simulations times of 100 ps and up to 500 atoms with acceptable accuracy and computational costs. This allows to work with explicit solvents if the considered cells and the adsorbed molecule(s) are small enough. As all the molecular modelling methods, AIMD simulations determine the total energy of the system and, hence, one can calculate adsorption energies at finite temperature. This is usually performed by averaging the internal energy of each system (surface alone, molecule alone, and surface with the adsorbed molecule) on total simulation times of 100 ps with however eliminating the first 10 ps of simulations, which are considered as a thermalization period.

General methods for liquid/solid adsorption modelling

A. Surfaces exposed Prior to the froth flotation process, a size reduction stage is performed to enable the transport of mineral particles by air bubbles. This crushing and milling stages cleave bonds existing between cations and anions in the mineral and generate new mineral surfaces. Minerals are cleaved following their usual cleavage planes, well described in the literature, which correspond to the surfaces with the lowest surface energy, i.e. the lowest surface density of broken bonds [START_REF] Whittaker | Cleavage energies of minerals[END_REF][START_REF] Gao | Mineral cleavage nature and surface energy: Anisotropic surface broken bonds consideration[END_REF] . Hence, a complete study has to be performed to investigate the most exposed surfaces for each mineral since all the adsorption processes will involve those surfaces [START_REF] Gao | Anisotropic surface broken bond properties and wettability of calcite and fluorite crystals[END_REF][START_REF] Hu | Anisotropic surface energies and adsorption behaviors of scheelite crystal[END_REF][START_REF] Gao | Anisotropic surface properties of calcite: A consideration of surface broken bonds[END_REF][START_REF] Gao | Surface broken bonds: An efficient way to assess the surface behaviour of fluorite[END_REF]. This investigation requires the calculation of surface energy for each different surface with accuracy and is usually performed by DFT or by HF methods.

In practice, different surfaces have to be generated from a bulk previously relaxed by first principles methods (DFT or HF), including the common cleavage planes that are usually well described in the crystallography area. Then, the different generated surfaces should be relaxed and the surface energy (in J.m -2 ) is calculated using:

()*+ = ( ,-+ . ( ,- . -) / * -) /

22

(5)

Where Eslab and Ebulk are the energies of the surface slab and the bulk cell, respectively, Nslab and Nbulk are the number of atoms contained in the slab and bulk unit cells, respectively, and A the slab area.

After the milling/crushing stages, the surface exposure follows a probability law that can be related to the surface energy differences between all the surfaces. However, at the moment, molecular modelling of the adsorption of flotation reagents cannot be performed on too many different surfaces since it would represent high computational costs. Hence, an assessment has to be done on the main exposed surface(s), which is/are selected for the following adsorption studies. The surface behaviour determination should constitute the first work to perform: for example, ab initio studies confirmed successfully that the (104) surface is the most exposed for calcite (CaCO3), a calcium carbonate [START_REF] Gao | Anisotropic surface broken bond properties and wettability of calcite and fluorite crystals[END_REF][START_REF] Gao | Anisotropic surface properties of calcite: A consideration of surface broken bonds[END_REF], the (001) and ( 112) for scheelite (CaWO4), a calcium tungstate [START_REF] Cooper | A combined ab initio and atomistic simulation study of the surface and interfacial structures and energies of hydrated scheelite: introducing a CaWO4 potential model[END_REF][START_REF] Gao | Surface energies and appearances of commonly exposed surfaces of scheelite crystal[END_REF], and the ( 001) and ( 210) for barite (BaSO4), a barium sulfate [START_REF] Bittarello | Ab Initio Calculations of the Main Crystal Surfaces of Baryte[END_REF].

B. Hydration

The froth flotation process and, most of the time, the prior milling stage are performed in aqueous conditions. It induces that the surfaces generated during the milling stage are exposed to water molecules before being in contact with flotation reagents. Hence, these later, when added to the pulp, interact with hydrated surfaces, i.e. surfaces on which water molecules are already adsorbed. The 4 systems should be relaxed with their energy calculated, summed for each member of the equation (left and right), and compared with each other to assess the preference of the anion substitution. Finally, the point 4 should be investigated by increasing the surface coverage: from a lone water molecule, one should add sequentially water molecules to reach a 100% coverage prior to add a second and a third water molecules layer. The final step is to conduct a simulation with the cell full of water molecules with a calculated density of 1 kg.dm -3 . The 4 above-mentioned possible hydration mechanisms can be investigated either at 0 K or at a finite temperature using DFT (static or AIMD, respectively). Besides, the points 1 and 4 can be addressed by CMD simulations since they involve no reactivity (bond creation/breakage).

The hydration mechanisms of widely known mineral surfaces have been intensively investigated over the past fifteen years by atomistic simulations. Various sulphides were studied by DFT simulations (M. Badawi et al., 2011b;J. Chen et al., 2014;[START_REF] Haider | Catalytic Dissociation of Water on the (001), (011), and (111) Surfaces of Violarite, FeNi 2 S 4 : A DFT-D2 Study[END_REF]Long et al., 2016b;[START_REF] Li | DFT Simulation on Interaction of H 2 O Molecules with ZnS and Cu-Activated Surfaces[END_REF] to bring clarity in the hydration mechanisms, mostly accompanied by oxidation phenomena. In addition, oxide minerals were deeply investigated since they represent an important source for non-base metals and mineral commodities. The hydration mechanisms of calcium/magnesium carbonates were described by CMD simulations [START_REF] Perry | Molecular models of a hydrated calcite mineral surface[END_REF][START_REF] Wolthers | Calcite surface structure and reactivity: molecular dynamics simulations and macroscopic surface modelling of the calcite-water interface[END_REF][START_REF] Fenter | Is the Calcite-Water Interface Understood? Direct Comparisons of Molecular Dynamics Simulations with Specular X-ray Reflectivity Data[END_REF], by the HF method [START_REF] Villegas-Jiménez | Theoretical Insights into the Hydrated (10.4) Calcite Surface: Structure, Energetics, and Bonding Relationships[END_REF]), and, mainly, by DFT calculations (de Leeuw and Parker, 1997[START_REF] Sun | COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase ApplicationsOverview with Details on Alkane and Benzene Compounds[END_REF][START_REF] Escamilla-Roa | Adsorption of Molecules onto (101̅ 4) Dolomite Surface: An Application of Computational Studies for Microcalorimetry[END_REF][START_REF] Lopez-Berganza | Ab Initio Studies of Calcium Carbonate Hydration[END_REF][START_REF] Goverapet Srinivasan | A comparative study of surface energies and water adsorption on Ce-bastnäsite, La-bastnäsite, and calcite via density functional theory and water adsorption calorimetry[END_REF] including AIMD simulations [START_REF] Ghatee | Bonding, structural and thermodynamic analysis of dissociative adsorption of H3O+ ion onto calcite (101¯4)$$ \left(10\overline{1}4\right) $$ surface: CPMD and DFT calculations[END_REF]. Other sparingly soluble salts were also deeply investigated with atomistic simulations to consider hydrated surfaces for the flotation reagents adsorption, such as scheelite [START_REF] Cooper | A combined ab initio and atomistic simulation study of the surface and interfacial structures and energies of hydrated scheelite: introducing a CaWO4 potential model[END_REF]de Leeuw and Cooper, 2003a), fluorite (CaF2) [START_REF] De Leeuw | Density functional theory calculations of adsorption of water at calcium oxide and calcium fluoride surfaces[END_REF][START_REF] Khatib | Water orientation and hydrogen-bond structure at the fluorite/water interface[END_REF]Foucaud et al., 2018a), apatite [Ca5(PO4)3(OH,F,Cl)] [START_REF] Zahn | Computational study of interfaces between hydroxyapatite and water[END_REF][START_REF] Pareek | Structure of Interfacial Water on Fluorapatite (100) Surface[END_REF][START_REF] Pareek | Competitive Adsorption of Glycine and Water on the Fluorapatite (100) Surface[END_REF][START_REF] Ulian | Probing the interaction of (001) carbonated hydroxylapatite surfaces with water: a density functional investigation[END_REF], and rare-earth-elements-bearing minerals [START_REF] Srinivasan | Crystal Structures, Surface Stability, and Water Adsorption Energies of La-Bastnäsite via Density Functional Theory and Experimental Studies[END_REF][START_REF] Goverapet Srinivasan | A comparative study of surface energies and water adsorption on Ce-bastnäsite, La-bastnäsite, and calcite via density functional theory and water adsorption calorimetry[END_REF], 2017;[START_REF] Stack | Mineral-Water Interface Structure of Xenotime (YPO 4 ) {100}[END_REF]. Metallic oxides/hydroxides such as rutile (TiO2) [START_REF] Heydari | Role of oxygen vacancy in the adsorption and dissociation of the water molecule on the surfaces of pure and Ni-doped rutile (110): a periodic full-potential DFT study[END_REF], manganite [MnO(OH)] [START_REF] Xia | The Manganite-Water Interface[END_REF], corundum (Al2O3) [START_REF] Janeček | Influence of Hydrogen Bonding on the Structure of the (001) Corundum-Water Interface. Density Functional Theory Calculations and Monte Carlo Simulations[END_REF], pyrochlore [START_REF] Bjørheim | Hydration thermodynamics of pyrochlore structured oxides from TG and first principles calculations[END_REF], goethite [FeO(OH)] [START_REF] Aquino | Quantum Chemical Adsorption Studies on the (110) Surface of the Mineral Goethite[END_REF][START_REF] De Leeuw | Surface simulation studies of the hydration of white rust Fe(OH)2, goethite α-FeO(OH) and hematite α-Fe2O3[END_REF]Y. Chen et al., 2017), and hematite (Fe2O3) (de [START_REF] De Leeuw | Surface simulation studies of the hydration of white rust Fe(OH)2, goethite α-FeO(OH) and hematite α-Fe2O3[END_REF][START_REF] Souvi | A DFT study of the hematite surface state in the presence of H2, H2O and O2[END_REF] were also studied since they are known to form hydroxylated species at their surfaces in the presence of water molecules. In particular, some surfaces can be hydroxylated and hydrated at the same time, with specific terminations [START_REF] Souvi | Influence of water, dihydrogen and dioxygen on the stability of the Cr 2 O 3 surface: A first-principles investigation[END_REF]. The same phenomenon occurs for silicates with various crystallographic structures, which constitute the most common gangue minerals in ores. Hence, their consideration in atomistic studies enables their better rejection by optimising the flotation reagents. The hydration mechanisms of many silicates, e.g. olivine [START_REF] Kerisit | Water and carbon dioxide adsorption at olivine surfaces[END_REF][START_REF] Prigiobbe | Interaction between Olivine and Water Based on Density Functional Theory Calculations[END_REF][START_REF] Liu | Water adsorption on olivine(010) surfaces: Effect of alkali and transition metal cation doping[END_REF] or various phyllosilicates [START_REF] Wang | Structure, Energetics, and Dynamics of Water Adsorbed on the Muscovite (001) Surface: A Molecular Dynamics Simulation[END_REF][START_REF] Wungu | First principles calculation on the adsorption of water on lithium-montmorillonite (Li-MMT)[END_REF][START_REF] Peng | A periodic DFT study of adsorption of water on sodiummontmorillonite (001) basal and (010) edge surface[END_REF], have been investigated with however a high number of studies focussing on quartz (SiO2) [START_REF] Rignanese | First-principles molecular-dynamics investigation of the hydration mechanisms of the (0001) α-quartz surface[END_REF][START_REF] Du | Molecular dynamics simulations of hydration, dissolution and nucleation processes at the α-quartz (0001) surface in liquid water[END_REF]Skelton et al., 2011;[START_REF] Wang | Structural and Electronic Properties of Different Terminations for Quartz (001) Surfaces as Well as Water Molecule Adsorption on It: A First-Principles Study[END_REF]. Interestingly, Skelton and co-workers compared different force fields for CMD simulations, used AIMD calculations to determine the best force field, and compared the water structure at the vicinity of the surface with experimental data. Along with hydration, interactions between electrolytes in the aqueous phase and mineral surfaces can be described by molecular modelling methods, mainly to investigate the depressing or activating effect of some ions such as phosphate anions [START_REF] Ahmed | Infrared spectroscopic characterization of phosphate binding at the goethite-water interface[END_REF], metallic cations [START_REF] Liu | DFT study of ethyl xanthate interaction with sphalerite (110) surface in the absence and presence of copper[END_REF][START_REF] Pfeiffer-Laplaud | Electrolytes at the Hydroxylated (0001) α-Quartz/Water Interface: Location and Structural Effects on Interfacial Silanols by DFT-Based MD[END_REF][START_REF] Guichen | An atomic scale investigation of the adsorption of sodium oleate on Ca2+ activated quartz surface[END_REF]. Overall, an accurate determination of the hydration state of a mineral is a crucial prerequisite for the investigation of reagents adsorption on a realistic surface. mechanisms of sulphides minerals (Fig. 3) such as galena (J. Chen et al., 2017), sphalerite [START_REF] Chen | A first-principle study of the effect of vacancy defects and impurities on the adsorption of O2 on sphalerite surfaces[END_REF], pyrite [START_REF] Sun | Oxygen adsorption on pyrite (100) surface by density functional theory[END_REF], and chalcopyrite [START_REF] Xiong | Oxidation mechanism of chalcopyrite revealed by X-ray photoelectron spectroscopy and first principles studies[END_REF], since oxidation reactions are known to affect the collector adsorption. Most of them combined DFT with spectroscopic and other experimental techniques to bring insights from ab initio molecular modelling or to validate the developed models. Besides, the adsorption of the most widely used collectors for sulphides (xanthates, thiophosphate, carbamate…) have been investigated by DFT for pyrite [START_REF] Hung | Density-functional theory studies of xanthate adsorption on the pyrite FeS2(110) and (111) surfaces[END_REF][START_REF] Hung | Density-functional theory of xanthate adsorption on the pyrite FeS 2 (100) surface[END_REF], for galena (J.-H. Chen et al., 2014;Long et al., 2016b), chalcopyrite [START_REF] Jiao | Adsorption mechanism of 2-mercaptobenzothiazole on chalcopyrite and sphalerite surfaces: Ab initio and spectroscopy studies[END_REF][START_REF] Zhao | Synthesis of Novel Ether Thionocarbamates and Study on Their Flotation Performance for Chalcopyrite[END_REF][START_REF] Ma | Structural Modification of Xanthate Collectors To Enhance the Flotation Selectivity of Chalcopyrite[END_REF][START_REF] Sarvaramini | Understanding the interactions of thiophosphorus collectors with chalcopyrite through DFT simulation[END_REF], and sphalerite [START_REF] Liu | DFT computation of Cu adsorption on the S atoms of sphalerite (110) surface[END_REF][START_REF] Liu | DFT study of ethyl xanthate interaction with sphalerite (110) surface in the absence and presence of copper[END_REF][START_REF] Jiao | Adsorption mechanism of 2-mercaptobenzothiazole on chalcopyrite and sphalerite surfaces: Ab initio and spectroscopy studies[END_REF]Long et al., 2016a;Sarvaramini et al., 2016b;[START_REF] Liu | DFT study of SDD and BX adsorption on sphalerite (1 1 0) surface in the absence and presence of water molecules[END_REF] with some concerns on the activation of sulphides by metallic ions such as Pb 2+ (Sarvaramini et al., 2016b) or Cu 2+ for sphalerite [START_REF] Liu | DFT study of ethyl xanthate interaction with sphalerite (110) surface in the absence and presence of copper[END_REF]. Since all the sulphides collectors are anionic, the use of DFT is compulsory to describe accurately the chemical bonding between the surface cations and the collectors anions (O or S). Moreover, the low electronegativity difference existing between sulphur and cations comprised in sulphides (Cu, Fe, Pb, Zn, Ni…) induce a strong-covalent component for their bond, which makes the use of DFT mandatory for such systems.

On silicate minerals

Few studies exist which focus on the adsorption of flotation reagents onto silicates since these latter are usually gangue minerals that have to be rejected and, thus, not collected in flotation. However, reverse flotation processes, in which silicates are collected, have been increasingly used over the past decades to purify the final metallic concentrate after the main separation, e.g. for iron ores [START_REF] Araujo | Reagents in iron ores flotation[END_REF][START_REF] Veloso | Investigation of the interaction mechanism of depressants in the reverse cationic flotation of complex iron ores[END_REF]. Hence, cationic collectors, mostly amines, adsorption have been studied

by CMD and DFT simulations on quartz [START_REF] Huang | Investigations on reverse cationic flotation of iron ore by using a Gemini surfactant: Ethane-1,2-bis(dimethyl-dodecylammonium bromide)[END_REF]Rath et al., 2014a;[START_REF] Liu | Quantum chemical calculations and molecular dynamics simulations of amine collector adsorption on quartz (0 0 1) surface in the aqueous solution[END_REF] which was in accordance with the generated experimental data (flotation tests and microcalorimetry results). The efficiency of new formulations such as ether and ester derivatives of primary aliphatic amines could be investigated, predicted and discussed based on DFT calculations (Rath et al., 2014a).

Besides, some silicates are collected in flotation since they represent either a metal-bearing mineral or an industrial mineral. Hence, the adsorption of amines onto such minerals was investigated by DFT calculations, e.g. on muscovite [START_REF] Xu | Adsorption of cationic collectors and water on muscovite (001) surface: A molecular dynamics simulation study[END_REF][START_REF] Wang | Mechanism of separating muscovite and quartz by flotation[END_REF] or on kaolinite [START_REF] Xia | Electron bandstructure of kaolinite and its mechanism of flotation using dodecylamine as collector[END_REF].

Moreover, the adsorption of anionic collectors, mostly fatty acids, was investigated by molecular modelling for quartz [START_REF] Li | Molecular dynamics simulation of siderite-hematite-quartz flotation with sodium oleate[END_REF] that can be activated by alkali-earth cations [START_REF] Guichen | An atomic scale investigation of the adsorption of sodium oleate on Ca2+ activated quartz surface[END_REF], but mostly for various valuable silicates that have a significant affinity for fatty acids such as feldspars (Xu et al., 2017a(Xu et al., , 2017b) ) or spodumene [START_REF] Fushun | Investigation on different behavior and mechanism of Ca2+, Fe3+ adsorption on spodumene surface[END_REF][START_REF] He | First-principles theory on electronic structure and floatability of spodumene[END_REF][START_REF] Yu | Role of oleic acid ionic-molecular complexes in the flotation of spodumene[END_REF]Xu et al., 2017a;[START_REF] Zhu | States of coadsorption for oleate and dodecylamine at selected spodumene surfaces[END_REF]. Since silicates are usually gangue minerals, the molecular modelling of depressants adsorption onto their surfaces has been investigated by DFT [START_REF] Han | DFT simulation of the adsorption of sodium silicate species on kaolinite surfaces[END_REF] although very few studies are currently reported.

On oxide and fluoride minerals

Oxide minerals were also strongly investigated by atomistic simulations to gain understanding in the flotation reagents adsorption mechanisms. As for sulphides, anionic collectors are widely used for oxides and require the use of DFT to describe correctly chemisorption [START_REF] Foucaud | Comment on: Effects of crystal chemistry on sodium oleate adsorption on fluorite surface investigated by molecular dynamics simulation[END_REF], although the covalent component is lower for oxides: except silicates, most of them include alkali, alkali-earth, or rare-earth elements as cations, which display higher electronegativity difference with oxygen compared to transition metals and sulphur. Otherwise, anionic collectors can be studied under their neutral, i.e. protonated, form to avoid the problem of reactivity. In this case, researchers have demonstrated that a physisorption occurs, based on electrostatic interactions between the collector polar group and the surface cation. Actually, one of the main limitation using DFT periodic codes is that the system has to be electrically neutral, inducing that either an extra-electron or a counter-ion that will give an electron to the anionic reagent must be added to the system.

The adsorption of anionic collectors has been mainly investigated on semi-soluble salt minerals since they constitute sources for some critical raw materials. For instance, researchers have studied the adsorption mechanisms of anionic collectors, mostly carboxylate and hydroxamate, by DFT onto rareearth minerals (Sarvaramini et al., 2016a;[START_REF] Espiritu | The effect of dissolved mineral species on bastnäsite, monazite and dolomite flotation using benzohydroxamate collector[END_REF][START_REF] Espiritu | Flotation behavior and electronic simulations of rare earth minerals in the presence of dolomite supernatant using sodium oleate collector[END_REF], onto scheelite [START_REF] Cooper | A Computer Modeling Study of the Competitive Adsorption of Water and Organic Surfactants at Surfaces of the Mineral Scheelite[END_REF][START_REF] Zhao | The DFT study of cyclohexyl hydroxamic acid as a collector in scheelite flotation[END_REF][START_REF] Yin | Structure-activity relationship and mechanisms of reagents used in scheelite flotation[END_REF], cassiterite (SnO2) [START_REF] Gong | In Situ Investigation of the Adsorption of Styrene Phosphonic Acid on Cassiterite (110) Surface by Molecular Modeling[END_REF][START_REF] Liu | New Insights into the Adsorption of Oleate on Cassiterite: A DFT Study[END_REF][START_REF] Tian | Novel insights into adsorption mechanism of benzohydroxamic acid on lead (II)-activated cassiterite surface: An integrated experimental and computational study[END_REF], apatite [START_REF] Mkhonto | A computer modelling study of the interaction of organic adsorbates with fluorapatite surfaces[END_REF][START_REF] Xie | Effects of structure of fatty acid collectors on the adsorption of fluorapatite (0 0 1) surface: A first-principles calculations[END_REF], hematite [START_REF] Chernyshova | Adsorption of Fatty Acids on Iron (Hydr)oxides from Aqueous Solutions[END_REF]Rath et al., 2014b), and fluorite (de Leeuw and Cooper, 2003b;Foucaud et al., 2018b). In addition, AIMD simulations can be performed to assess the influence of the temperature on the adsorption mechanisms (bond lengths and angles, molecule steric configuration, adsorption energy, adsorption geometry…). We demonstrated that the adsorption of fatty acids under their anionic form was impacted by the temperature. Finally, water molecules should be added under the form of an explicit solvent to consider the effect of the liquid phase on all the studied parameters. Remarkably, we reported that the fatty acids do not adsorb under the same geometrical configuration in 'static' DFT and AIMD (300 K) simulations (Foucaud et al., 2018b).

Conclusion

The froth flotation process is led by the adsorption of various reagents, organic or mineral, at the liquid/solid interface, with the objective of selectively rendering the target mineral(s) hydrophobic to recover it in the froth phase. Collectors, which contain a polar group and a non-polar aliphatic chain, can be adapted in terms of chain length, unsaturation, and ramification, as well as functionalized polar group. Moreover, all reagents can be combined to improve the flotation performances (selectivity or target mineral(s) recovery). All these optimisations are difficult to investigate by experimental methods and molecular modelling is a powerful tool to gain understanding in the adsorption mechanisms of flotation reagents at the mineral surfaces. Two main methods are widely used, forcefields-based (CMD) and ab initio simulations (DFT), with significant differences in accuracy, considered phenomena, computational costs, system sizes and simulation times. One should adapt the atomistic method to the flotation system, mostly to the collector. These theoretical techniques can be used as predictive methods to improve the flotation performances which, however, requires to characterize as well as possible the hydration state, the depressants adsorption, and the collector adsorption. The use of molecular modelling in flotation is tremendously increasing and will be, in the future, a crucial tool to attain the flotation performances required for the processing of low-grades fine-grained complex ores by predictive calculations.

Fig. 1 .

 1 Fig. 1. Number of research articles per year over the past fifteen years including the words "flotation", "molecular", and "simulation" (a) and including the words "flotation" and "DFT" (b) from Web of Science.

  supply a relatively accurate description of the physisorption phenomena, which represent a significant part of the molecular mechanisms involved in the flotation process. Moreover, reactive force fields such as the ReaxFF (van Duin et al., 2001) have been developed over the past decades to include chemical reactivity. They are based on high accurate ab initio calculations and allow continuous bond formation/breakage on, nonetheless, a restricted number of atom types at the moment (van Duin et al.,

  Therefore, the molecular modelling of the adsorption of flotation reagents has to be preceded by a deep study of the surface hydration mechanisms. Overall, several molecular mechanisms are at play at the mineral/water interface: 1. Adsorption of molecular water 2. Adsorption of dissociated water, leading to a hydroxylation of surface cations 3. Substitution of a surface anion by a hydroxyl group (HO -) 4. Influence of the surface on several water layers at the interface To consider mechanisms 1 and 2, the adsorption energy of a lone water molecule should be calculated and compared with the adsorption energy of a dissociated water molecule, where the H atom is set on a surface anion while the HO group is placed on a surface cation. The point 3 should be investigated by considering the following reaction, with A being the surface anion: Surface-A + H2O = Surface-OH + HA (6)

Fig. 2 .

 2 Fig. 2. Snapshots of MD simulations using force fields (left) or DFT (right) to investigate the hydration mechanisms of calcite (left) and fluorite (right) surfaces. Adapted from (Perry et al., 2007) (left) and from (Foucaud et al., 2018a) (right).

Fig. 3 .

 3 Fig. 3. Adsorption of solvated diisobutyl dithiophosphinate on sphalerite surface: most stable adsorbed configuration (left), non-adsorbed solvated collector (right), adapted from (Sarvaramini et al., 2016b).

Fig. 4 .

 4 Fig. 4. Snapshots of AIMD simulations (300 K) presenting the configuration differences of the polar head and the non-polar chain of an octanoate molecule between the adsorption in vacuum (left) and in explicit water molecules (b), adapted from(Foucaud et al., 2018b).

Acknowledgements

The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N o. 641650 for the FAME project. We also want to acknowledge the support of Labex Ressoures 21, supported by the French National Research Agency through the national programme "Investissements d'Avenir" with reference ANR-10-LABX-21-LABEX RESSOURCES 21.