Marie-Caroline Prima
email: marie-caroline.prima@univ-grenoble-alpes.fr

Mathieu Garel

Pascal Marchand

Luca Börger

Florian Barnier

Combined effects of landscape fragmentation and sampling frequency of movement data on the assessment of landscape connectivity

Keywords: Alpine ibex, Dead-reckoning, Distance-based networks, Landscape fragmentation, Minimum planar graph, Movement behaviour, Sampling frequency, Scale-free, Small-world, Spatial network theory

Background -

Network theory is largely applied in real-world systems to assess landscape connectivity using empirical or theoretical networks. Empirical networks are usually built from discontinuous individual movement trajectories without knowing the effect of relocation frequency on the assessment of landscape connectivity while theoretical networks generally rely on simple movement rules. We investigated the combined effects of relocation sampling frequency and landscape fragmentation on the assessment of landscape connectivity using simulated trajectories and empirical high-resolution (1 Hz) trajectories of Alpine ibex (Capra ibex). We also quanti ed the capacity of commonly used theoretical networks to accurately predict landcape connectivity from multiple movement processes.

Methods -

We simulated forager trajectories from continuous correlated biased random walks in simulated landscapes with three levels of landscape fragmentation. High-resolution ibex trajectories were reconstructed using GPS-enabled multi-sensor biologging data and the dead-reckoning technique. For both simulated and empirical trajectories, we generated spatial networks from regularly resampled trajectories and assessed changes in their topology and information loss depending on the resampling frequency and landscape fragmentation. We nally built commonly used theoretical networks in the same landscapes and compared their predictions to actual connectivity.

Results -

We demonstrated that an accurate assessment of landscape connectivity can be severely hampered (e.g., up to 66% of undetected visited patches and 29% of spurious links) when the relocation frequency is too coarse compared to the temporal dynamics of animal movement. However, the level of landscape fragmentation and underlying movement processes can both mitigate the effect of relocation sampling frequency. We also showed that network topologies emerging from different movement behaviours and a wide range of landscape fragmentation were complex, and that commonly used theoretical networks accurately predicted only 30-50% of landscape connectivity in such environments.

Conclusions -

Very high-resolution trajectories were generally necessary to accurately identify complex network topologies and avoid the generation of spurious information on landscape connectivity. New technologies providing such high-resolution datasets over long period should thus grow in the movement ecology sphere. In addition, commonly used theoretical models should be applied with caution to the study of landscape connectivity in real-world systems as they did not perform well as predictive tools.

Background

Studying landscape connectivity is crucial for better understanding many ecological processes including animal foraging behaviour [1], population distribution [2,3], gene or pathogen ow [4], migration behaviour [5], species interactions [6,7], or metapopulation persistance [8]. Analysis of landscape connectivity can also be useful to identify critical areas favouring movements [9][START_REF] Belote | Examining local and regional ecological connectivity throughout North America[END_REF][START_REF] Chauveau | Identifying the environmental drivers of corridors and predicting connectivity between seasonal ranges in multiple populations of Alpine ibex (Capra ibex) as tools for conserving migration[END_REF] or to evaluate and compare the effect of land planning actions (e.g., site restoration) [START_REF] Lookingbill | Using landscape analysis to evaluate ecological impacts of battle eld restoration[END_REF][START_REF] Bergsten | To model the landscape as a network: A practitioner's perspective[END_REF]. Consequently, an accurate assessment and prediction of landscape connectivity is needed to avoid spurious understanding of multiple theoretical and applied landscape connectivity-related processes.

Network (or graph) theory has largely been used in multiple taxa to model and predict connectivity in fragmented landscapes [START_REF] Urban | Landscape connectivity: A graph-theoretic perspective[END_REF][START_REF] O'brien | Testing the importance of spatial con guration of winter habitat for woodland caribou: An application of graph theory[END_REF][16][START_REF] Wittemyer | Graph theory illustrates spatial and temporal features that structure elephant rest locations and re ect risk perception[END_REF][18]. In such landscapes, habitat patches are represented using nodes while observed or predicted movements among patches are represented using links [START_REF] Fall | Spatial graphs: principles and applications for habitat connectivity[END_REF]. Links can either mirror the Euclidean distance between patches (i.e., straight lines) [START_REF] Minor | A graph-theory framework for evaluating landscape connectivity and conservation planning[END_REF] or account for matrix composition among patches (i.e., least cost path) [6]. The popularity of spatial networks relies on their easy implementation and the large amount of connectivity indexes that can be derived from a network providing quantitative information on landscape connectivity at different levels (i.e., node, group of nodes or network level) [START_REF] Saura | A new habitat availability index to integrate connectivity in landscape conservation planning: Comparison with existing indices and application to a case study[END_REF][START_REF] Ray Eld | Connectivity for conservation: a framework to classify network measures[END_REF][START_REF] Martensen | Spatio-temporal connectivity: assessing the amount of reachable habitat in dynamic landscapes[END_REF]. Their adequatedness in real landscapes for correctly inferring ecological processes is, however, less immediately clear.

A landscape can be described using different network topologies including simple planar graphs (e.g., minimum spanning tree or minimum planar graph, MPG) [START_REF] Reunanen | Spatial graphs as templates for habitat networks in boreal landscapes[END_REF] and more complex ones (e.g., scale-free or small-world networks) [START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF]. They result in different patterns of connectivity within the graph (Fig. 1A) [26]. For example, a MPG assumes that individuals move in a stepping stone fashion among resource patches, such that links in a MPG never cross (i.e., no shortcut). By contrast, in a scale-free network, few nodes (thereafter called hubs) are highly connected to the rest of the patches (thereafter called peripheral nodes) that have few connections [START_REF] Barabasi | Emergence of scaling in random networks[END_REF]. Consequently, a scale-free network is highly sensitive to the removal of hubs but resistant to random removal of nodes, whereas a MPG shows similar effects of a disturbance independently of its spatial location [3,28]. Network topology also has implications for processes occurring in spatial networks such that a disease would not spread much if it appears in a peripheral node while it would spread faster and on a larger extent if appearing in a hub or in a network having shortcuts among patches (such as in small-world networks) [START_REF] Minor | A graph-theory framework for evaluating landscape connectivity and conservation planning[END_REF]. An accurate representation of actual landscape connectivity is thus of paramount importance to correctly identify network topology and consequently, to understand and predict many landscape connectivity-related processes.

Application of spatial network theory to real-world landscapes generally relies on the modelling of either theoretical networks (i.e., commonly a MPG built from the distribution of resource patches [START_REF] O'brien | Testing the importance of spatial con guration of winter habitat for woodland caribou: An application of graph theory[END_REF]6,[START_REF] Leblond | Science-informed policy decisions lead to the creation of a protected area for a wide-ranging species at risk[END_REF], or a distance-based network built from both distribution of resource patches and empirically-estimated or expert-based dispersal distances [START_REF] Minor | A multiscale network analysis of protected-area connectivity for mammals in the United States[END_REF]9,4]) or empirical networks [START_REF] Wittemyer | Graph theory illustrates spatial and temporal features that structure elephant rest locations and re ect risk perception[END_REF]18]. Empirical graphs are generated by overlapping trajectories of remote-tracked individuals with resource patches to identify network links [18,[START_REF] Fox | Herbivores in a small world: network theory highlights vulnerability in the function of herbivory on coral reefs[END_REF]. With the development of Global Positioning System (GPS) technology, many free-ranging species are now tracked over long periods of time [START_REF] Cagnacci | Animal ecology meets GPS-based radiotelemetry: a perfect storm of opportunities and challenges[END_REF][START_REF] Wilson | Identifying polar bear resource selection patterns to inform offshore development in a dynamic and changing Arctic[END_REF]. However, GPS technology shows weight constraints that limit battery capacities, generally resulting in a trade-off between relocation frequency and tracking duration depending on scienti c questions [START_REF] Hofman | Right on track? Performance of satellite telemetry in terrestrial wildlife research[END_REF]. For examples, empirical networks have been built from GPS relocations collected every hour in plains bison (Bison bison bison) [3] or in african elephants (Loxodonta africana) [START_REF] Wittemyer | Graph theory illustrates spatial and temporal features that structure elephant rest locations and re ect risk perception[END_REF], every two hours in woodland caribou (Rangifer tarandus caribou) [START_REF] Prima | A landscape experiment of spatial network robustness and space-use reorganization following habitat fragmentation[END_REF], every four hours in brown bear (Ursus arctos) [36] or even every 24 hours in several species of sea turtle [18]. However, the potential effect of relocation frequency on observed patterns of connectivity remains unclear. Indeed, discontinuous trajectories like the one generated from GPS monitoring could result in inaccurate estimations of inter-patch movements and, for example, the detection of arti cial shortcuts due to missed stepping stone-like movements. Consequently, relocation frequency could impact the assessment of connectivity in a sense that networks are perceived as more complex, while being actually simpler (e.g., arti cial shortcuts could generate a small-world topology). On one hand, if networks are actually simpler than detected, it assures the relevant use of simple theoretical networks to predict landscape connectivity (i.e., such as the MPG). On the other hand, if networks are actually complex, there is an urgent need to develop theoretical network models that t with complex functional patterns of connectivity.

In patchy landscapes, the level of habitat fragmentation could also impact observed patterns of landscape connectivity from remote-tracked individuals. Indeed, habitats show different levels of fragmentation starting from simple patch dissection to patch attrition with intermediate patterns such as patch shrinkage [START_REF] Jaeger | Landscape division, splitting index, and effective mesh size: new measures of landscape fragmentation[END_REF], resulting in habitat patches having different size, organisation and composition [38]. While patches can provide different resources to animals (e.g., food, shelter), their use (e.g., residency time, number of visits) can vary according to patch characteristics [3,6,[START_REF] Prima | A landscape experiment of spatial network robustness and space-use reorganization following habitat fragmentation[END_REF][START_REF] Cresswell | The effect of patch size and separation on bumblebee foraging in oilseed rape: implications for gene ow[END_REF]. For example, plains bison and woodland caribou were shown to stay longer in larger and highly pro table patches [START_REF] Prima | A landscape experiment of spatial network robustness and space-use reorganization following habitat fragmentation[END_REF][START_REF] Courant | Search e ciency of free-ranging plains bison for optimal food items[END_REF]. Such variations in residency time within habitat patches could result in missed inter-patch movements (e.g., stepping-stones), notably when the sampling schedule is too coarse compared to the temporal dynamics of animal movement [START_REF] Saura | Stepping stones are crucial for species' long-distance dispersal and range expansion through habitat networks[END_REF]. While the interplay between landscape fragmentation and relocation frequency could have noteworthy effects on the assessment of landscape connectivity using networks, it has yet to be appraised.

Finally, individuals could also adjust their movements as a response to space use of congeners, competitors or predators [START_REF] Moorcroft | Mechanistic home range models capture spatial patterns and dynamics of coyote territories in Yellowstone[END_REF][START_REF] Nathan | A movement ecology paradigm for unifying organismal movement research[END_REF][START_REF] Laundré | Behavioral response races, predator-prey shell games, ecology of fear, and patch use of pumas and their ungulate prey[END_REF][START_REF] Courbin | Reactive responses of zebras to lion encounters shape their predator-prey space game at large scale[END_REF], such that landscape connectivity could result in different patterns according to the movement process at stake. For example, individuals can have limited access to the landscape when conspeci cs are territorial and consequently have to restrain their movement to their own territory [46]. Similarly, space use and movements can be modi ed by prey individuals in response to a threat (e.g., predator or human presence) [START_REF] Courbin | Reactive responses of zebras to lion encounters shape their predator-prey space game at large scale[END_REF][START_REF] Courbin | Interacting lethal and nonlethal human activities shape complex risk tolerance behaviors in a mountain herbivore[END_REF]. Consequently, we could expect network topology to be impacted by movement processes and, if so, these movement processes should be accounted for when building theoretical network models to predict connectivity in heterogeneous landscapes.

In this study, we address these outstanding methodological questions by investigating the combined effects of relocation sampling frequency and landscape fragmentation on the assessment of landscape connectivity using (i) simulated trajectories accounting for four movement processes (foraging, foraging + avoidance of an elusive predator, foraging + avoidance of a stalking predator, foraging and territoriality) in simulated habitats with variable levels of fragmentation and (ii) empirical high-resolution (1 Hz) trajectories of Alpine ibex (Capra ibex) tracked in the northern French Alps (Belledonne massif ; Fig. 1C). For both simulated and empirical trajectories, we also compared predictions from commonly used theoretical networks (i.e., MPG and distance-based graphs) to actual connectivity obtained from highresolution data.

Methods

Simulation study

Modelling of movement processes

We used a recently developed modelling framework to simulate individual trajectories in fragmented landscapes from continuous correlated biased random walks [48][START_REF] Bracis | Remembering the good and the bad: memory-based mediation of the food-safety trade-off in dynamic landscapes[END_REF][START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF]. This framework simulates an animal grazing across stationary resources that deplete and regenerate, based on three processes: consumption and regeneration of resources, resource memory of the grazers, and state-speci c biased correlated movement process [48]. The framework can also accommodate predator avoidance during movements by simulating predator appearance in the landscape and a resulting ying-behaviour by the grazer that memorizes an encounter location and avoids it for a parameterized period of time [START_REF] Bracis | Remembering the good and the bad: memory-based mediation of the food-safety trade-off in dynamic landscapes[END_REF]. The model can nally include multiple territorial foragers that regularly scent-mark their territory and avoid scent-marks of conspeci cs [START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF]. Detailed descriptions of foraging, predator avoidance and territoriality models are available in [48], [START_REF] Bracis | Remembering the good and the bad: memory-based mediation of the food-safety trade-off in dynamic landscapes[END_REF] and [START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF], respectively, such that we only provide a brief description of each modelled process in Appendix S1.

Modelling of landscape fragmentation

We generated three types of landscapes of 50×50 cells with different levels of fragmentation using an exponential variogram model. This model was characterized by four parameters including the nugget (i.e., variogram's intercept), the sill (i.e., variogram's asymptote), the range (r, i.e., distance beyond which variables are no longer autocorrelated) and the trend (t, i.e., average predicted value over the landscape).

Constant parameters of the model were: nugget = 0 and sill = 1. We set range and trend coe cients to obtain three levels of fragmentation: low (r = 5, t = 0.3), intermediate (r = 2, t = -0.5) and high (r = 2, t = -1.5) [START_REF] Deutsch | GSLIB: Geostatistical software library and user's guide[END_REF][START_REF] Bivand | Applied spatial data analysis with R[END_REF]. We produced 100 landscapes for each set of parameters to account for stochasticity. Following [48], negative values were truncated to 0 and landscapes were normalized to sum to one. Thus, cells with positive values re ected ressources of gradual quality while null values corresponded to non-ressource cells. We used the gstat package (v2.1-1) of the R software (v4.3.0) to produce the landscapes [START_REF] Core | R: A language and environment for statistical computing[END_REF][START_REF] Pebesma | Multivariable geostatistics in S: the gstat package[END_REF]. We then checked for differences in the level of fragmentation between simulated landscapes using four indexes of landscape fragmentation (see Appendix S2).

Model runs and construction of spatial networks

We ran each model (F: foraging-only, F + Pe: foraging and avoidance of an elusive predator, F + Ps: foraging and avoidance of a stalking predator, F + T: foraging and territoriality) in each simulated landscape resulting in 1200 simulations (100 landscapes of each type x 4 movement processes x 3 levels of landscape fragmentation). Continuous time models were implemented in Java, with time discretized with small regular intervals Δt approximating dt. For each simulation, we ran the model during 10 000 timesteps. As for the F + T model, territories have to emerge rst, we ran 20 000 time steps and discarded the rst 10 000 to only keep movement behaviour when territories were in place [START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF]. Model parameters for each simulated process were initialized based on [48][START_REF] Bracis | Remembering the good and the bad: memory-based mediation of the food-safety trade-off in dynamic landscapes[END_REF][START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF], and are reported in Table S1.

All of the subsequent analysis were performed using the R software (v4.3.0, [START_REF] Core | R: A language and environment for statistical computing[END_REF]). We then built a network for each complete simulated trajectory (i.e., 1200 networks). To do so, we identi ed all movements that occurred between two resource patches and used unique inter-patch movements to build the network (i.e., unweighted and undirectional network). Then, to mimic discontinuous trajectories like the one generated from GPS monitoring, we performed the same steps (i.e., from trajectories to spatial networks), but rst resampled each trajectory by joining one location out of two, one out of three, etc…, until one location out of fty, resulting in 4999, 3333, etc…, down to 199 movement steps, respectively.

Evaluation of simulated spatial network

Effects of landscape fragmentation and sampling frequency on network topology

Multiple network metrics are available to describe landscape connectivity and to distinguish between network types [START_REF] Ray Eld | Connectivity for conservation: a framework to classify network measures[END_REF]. Here, we focused on the degree distribution (i.e., the distribution of the number of links that each node has), the global clustering coe cient C (i.e., the number of connected triplets of nodes over the total number of triplets in the graph) and the characteristic path length L (i.e., the number of links in the shortest path between two nodes, averaged over all pairs of nodes) (Fig. 1B). These metrics allow evaluating whether a network topology is complex and similar to classic complex networks such as scalefree or small-world networks (Fig. 1A) [START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF]. In a scale-free network, the degree distribution follows a powerlaw distribution with scale parameter > 1 (i.e., the probability of a node to be connected to other nodes, ), re ecting the presence of hubs and peripheral nodes within the network [START_REF] Barabasi | Emergence of scaling in random networks[END_REF][START_REF] Lee | The role of network science in the study of anesthetic state transitions[END_REF]. In a small-world network, many triplets of nodes are connected resulting in a high clustering coe cient and some links provide long range connexion between distant nodes (e.g., shortcuts) leading to a short characteristic path length (i.e., fast traversability of the graph) [START_REF] Lee | The role of network science in the study of anesthetic state transitions[END_REF]56]. The ad hoc method to evaluate whether a network is small-world is to compare C and L to the clustering coe cient and characteristic path length of an associated random graph built from the same number of links and nodes than the one of interest [START_REF] Newman | The structure and function of complex networks[END_REF]58]. Notably, in a small-world network, the clustering coe cient is much larger than the clustering coe cient of its associated random graph and the characteristic path length is similar to the one of its associated random graph.

To evaluate the effects of landscape fragmentation, sampling frequency and movement process driving forager behaviour on the network topology, we rst extracted the node degree of all visited patches of each network. We then tted a linear mixed-effects model to the log-transformed node degree as a function of landscape fragmentation, the number of relocations used to build the network (correlated to sampling

α k P (X = k) ∝ k -α , α > 1, k ∈ N*
frequency, Fig. 1C) and the interaction of both. We included a random effect of landscape ID to account for autocorrelation in the degrees of patches coming from the same landscape. We ran a model for each simulated movement process (i.e., F, F + Pe, F + Ps and F + T). We also computed network degree distribution for each level of landscape fragmentation and each sampling frequency and tted the logtransformed frequency of node degrees to the log-transformed degrees using a linear model to estimate whether networks were scale-free (i.e., in such case estimated scale parameter should be higher than 1). We then extracted C and L of each network and built 100 random networks for each simulated network based on the same number of links and visited patches and compared their clustering coe cient (C R ) and characteristic path length (L R ) to C and L to assess whether the empirical network had the small-world property. More precisely, we followed [START_REF] Humphries | Network 'small-world-ness': A quantitative method for determining canonical network equivalence[END_REF] and calculated the ratio σ = (C/C R ) / (L/L R ) for each random graph meaning that we had one hundred calculation of σ for each simulated network. As these ratios should be higher than 1 if the simulated network is small-world [START_REF] Humphries | Network 'small-world-ness': A quantitative method for determining canonical network equivalence[END_REF], we de ned a simulated network to be small-world if min(σ) > 1. Finally, we calculated the proportion of simulated networks being small-world for each simulated movement process as a function of landscape fragmentation and the number of relocations used to build the network. We used the R packages igraph (v1. 4.3, [60]) to build the networks and calculate their connectivity indexes, and lme4 (v1. 1-33, [61]) and lmerTest (v3.1-3, [62]) to t the statistical models.

Adequacy with commonly used theoretical networks

We rst built the minimum planar graph in each simulated landscape using the R package grainscape [START_REF] Core | R: A language and environment for statistical computing[END_REF].

We also simulated two distance-based networks and a random graph in each landscape for each simulated movement process. Notably, we built distance-based networks using two thresholds for link length: one corresponding to the median length (L med ) of realized links obtained from unique inter-patch movements of the complete trajectory (DIST50), and one corresponding to the 95% quantile (L 95% ) of the distribution of realized link length (DIST95). Consequently, only patches that were closer than L med and L 95% were thus connected in DIST50 and DIST95, respectively. For the random graph, we randomly assigned the same number of realized links among patches of the landscape. We then used the Cohen's Kappa (κ) to assess the performance of theoretical spatial networks (MPG, DIST50, DIST95 and Random) to correctly predict connectivity emerging from different movement models in landscapes having various levels of fragmentation [63,64]. We speci cally calculated κ by comparing predicted links (0/1) of theoretical networks to realized links (0/1) from simulated movement processes in each landscape.

Empirical study -Movements of Alpine ibex in the Belledonne massif

Study area and Alpine ibex population monitoring

The empirical study took place in the Belledonne massif located in the northern French Alps (45° 13' N, 6°4 ' E). The tagged alpine ibex ranged over a 35km 2 area at an altitude around 2100 m within the core of the massif. Ibex are notably adapted to movements in steepy terrains, which provide them refuge from perceived predation risk, in particular the steepest rocky cliffs. However, cliffs generally provide few foraging resources, that are rather concentrated in Alpine grasslands [65,66], such that ibex move frequently among various habitat types to get both food and cover. In total, 10 male ibex (between 7 and 12 years old) were monitored in 2017 using Lotek (3300S and Litetrack models) and Vectronic (Vertex Plus model) GPS collars. Relocation frequency varied between 1 location per hour to 1 location every 2 hours and ibex were tracked most of the time nearly one year. Among these 10 ibex, 6 were additionally equipped also with Daily Diary multi-sensor biologgers (Wildbyte Technologies 2020), which include tri-axial accelerometer and tri-axial magnetometer sensors recording at 25 Hz/ 8 Hz, respectively. Data from these biologgers were combined with GPS data collected every one or two hours to reconstruct the movement trajectory every second using the dead-reckoning technique (DR) (hereafter GPS-enabled dead-reckoning) [67]. Given a starting point, DR uses the accelerometry and magnetometry data to calculate the speed and heading of an individual animal every second and thereby reconstruct the full high-frequency movement path [68]. To successfully use the DR method, in addition to correctly calibrating the sensors and correct for hard and soft iron bias (see [68]), it is important to recalibrate the reconstructed DR track as errors can accumulate and increase if not accounted for -this is typically done using GPS locations collected at a lower frequency, as detailed in [69]. We focused our analysis on the period for which we had very highresolution trajectories for all six ibex (i.e., June) resulting in a total of 3412 GPS relocations (341 locations on average per individual, range: 204-395) from the 10 GPS-collared ibex and 18 070 914 DR relocations (3 011 819 locations on average per individual, range: 3 010 683-3 012 356) from the 6 ibex that were also equipped with biologgers. Variations in the number of relocations between ibex from GPS-enabled dead-reckoning came from a few missing relocation estimates but as this phenomenon was marginal (i.e., less than 0.5% of the dataset) and the time difference between two successive relocations never exceeded 4 seconds, we considered missing location to be the same as the location immediately preceding it.

We used slope and aspect layers provided by the R package terra to describe terrain characteristics in the ibex range at a resolution of 13x13m. We also used description of habitats from the vegetation map provided by the Conservatoire botanique national alpin [START_REF] Sanz | Catalogue des végétations de l'Isère. Classi cation physionomique et phytosociologique avec clés de détermination[END_REF]. Notably, 19 habitats were identi ed within the ibex range from the vegetation map, which we reclassi ed into 8 classes including: closed forest, grassland, herbaceous -mineral, low ligneous, mineral, open forest, snow and other.

Identi cation of resource patches for Alpine ibex

To identify ibex resource patches, we rst quanti ed their habitat selection using a resource selection function (RSF) and geolocations of the 10 GPS-collared ibex. Model robustness was assessed using k-fold cross validation (see Appendices S3-S4) [START_REF] Boyce | Evaluating resource selection functions[END_REF]. We then calculated from the tted RSF a selection probability for each 13x13m pixel of the study area and retained pixels that had a probability above the 75% quantile of probability distribution. We then aggregated adjacent selected pixels to generate resource patches for ibex. We ltered patches on polygons being larger than 2000 m 2 to take into account the radius of GPS measurement error (i.e., π*25 2 m 2 ). Finally, once ibex resource patches were identi ed, we calculated the four indexes of landscape fragmentation (i.e., the aggregation, patch cohesion and division indexes, and the proportion of cells occupied by habitat patch, see Appendix S2) to identify to which fragmentation scenario from the simulated landscapes the ibex actual landscape was more similar. We used the R packages adehabitatHR (v0.4.21), sp (v1.6-1), stars (v0.6-1) and raster (v3.6-20) to perform all geographical information system work and lme4 (v1. and lmerTest (v3.1-3) to run the statistical analysis [START_REF] Bivand | Applied spatial data analysis with R[END_REF]61,62,[START_REF] Calenge | The package adehabitat for the R software: A tool for the analysis of space and habitat use by animals[END_REF][START_REF] Hijmans | Raster: Geographic data analysis and modeling[END_REF][START_REF] Bivand | Spatial data science: With applications in R[END_REF].

Construction of Alpine ibex spatial network at various sampling design

We used GPS-enabled dead-reckoning relocations obtained every second (1 Hz) for 6 ibex to rst build a very high-resolution spatial network. As for the simulation study, we identi ed all movement steps that occurred between two resource patches and used unique inter-patch moves to build the network. We then regularly resampled each trajectory to mimic various sampling designs and built the resulting spatial networks. Notably, we resampled each trajectory by joining one location out of 30 (30 seconds), 60 (1 minute), 300 (5 minutes), 600 (10 minutes), 1200 (20 minutes), 1800 (30 minutes), 3600 (1 hour), 7200 (2 hours) and 21600 (6 hours). We randomly selected the rst relocation to start the resampling for each ibex and each sampling design to generate stochasticity. For example, when relocations were sampled every 30 seconds, the relocation to start with was randomly selected between the rst 30 available relocations while for the sampling at 1 hour, the relocation to start with was selected between the rst 3600. In addition, for the 2 and 6 hours sampling design, we also shifted by one hour every day the sampling time to have a complete sample of all possible hours. For example, for the 6 hours sampling design, we sampled relocations day 1 at 00:00, 06:00, 12:00, 18:00 and 24:00, then day 2 at 01:00, 07:00, 13:00, 19:00, 01:00, then day 3 at 02:00, 08:00 and so on.

Evaluation of Alpine ibex spatial network

We performed the same analysis on ibex spatial networks as for the simulation study. We rst evaluated the effect of sampling frequency on node degree by tting a linear model to all log-transformed degrees as a function of the log-transformed number of relocations used to build the networks. We also assessed whether the networks had the scale-free and small-world properties by estimating the scale parameter of each degree distribution and by comparing their clustering coe cient and characteristic path length to the ones of random graphs, respectively (see section Effects of landscape fragmentation and sampling frequency on network topology). Secondly, we built the four commonly used theoretical spatial networks (i.e., MPG, DIST50, DIST95 and random graph) to predict connectivity among ibex resource patches. We restricted the predictions to patches that were encompassed within the 100% minimum convex polygon of the very high-resolution relocations (i.e., every second) of the 6 ibex. We then compared predictions of each theoretical network to the empirical observation of ibex spatial network derived from the very highresolution trajectory (1 Hz), using Cohen's Kappa.

In addition, we also calculated, for each network obtained from the resampled trajectories, the proportion of visited patches detected, the proportion of realized links detected and the proportion of spurious links detected. To do so, we used the spatial network built from the 1 Hz trajectory as the reference and extracted from it the lists of visited patches and realized links. We then evaluated whether these visited patches and links were found again in resampled networks. We de ned spurious links as links that were observed in resampled networks but missing in the reference network.

Results

Simulation study

Effects of landscape fragmentation and sampling frequency on simulated networks Simulated landscapes showed various fragmentation characteristics according to the four indexes of landscape fragmentation (described in Table S2). Each degree distribution calculated from 100 simulated networks decayed as a power function with scale parameter being larger than 1 (Table S3), indicating a strong heterogeneous pattern of connectivity among patches in all simulated networks. Few patches had many links (i.e., the hubs) and most of the remaining patches had few links, typical of scale-free networks (Figs. 2, S1 and S2). The scale-free property persisted even when sampling frequency decreased (i.e., less relocations) for all types of landscape fragmentation and all simulated movement processes (Fig. 2, Table S3). However, node degree depended on both sampling frequency and landscape fragmentation for all four simulated movement processes (Fig. 3). Node degree globally increased with the number of relocations but at a stronger rate in highly fragmented landscapes for the movement processes re ecting only foraging (i.e., F) or foraging and predator avoidance (i.e., F + Pe and F + Ps, Fig. 3), implying higher information loss on landscape connectivity in fragmented landscapes when sampling frequency gets coarser. This effect although dissipated when foragers had to constrain their movement within their own territory (Fig. 3, F + T).

The emergence of the small-world property was linked to both the simulated movement process and the level of landscape fragmentation (Fig. 4). Indeed, from the full trajectory (i.e., 10000 relocations), all simulated networks were small-world in highly and medium fragmented landscapes for the movement processes re ecting only foraging (i.e., F) or foraging and predator avoidance (i.e., F + Pe and F + Ps, Fig. 4). This proportion decreased though when landscapes had low fragmentation level or when territoriality was at stake in the movement process (F + T, Fig. 4). In addition, a decrease in the number of relocations generally led to a decrease in the proportion of small-world networks at a rate that depended on both the movement process and the fragmentation level (Fig. 4), meaning that a coarser sampling frequency impeded the detection of complex network topology.

Adequacy with commonly used theoretical spatial networks

Commonly used theoretical networks showed various accuracy to predict connectivity emerging from different movement processes in landscapes having various levels of fragmentation (Table 1) and never predicted correctly more than half of the landscape connectivity. Indeed, Cohen's Kappa varied between 0 and 0.49 with lowest values being the random graph and highest values the DIST50 in all four simulated movement processes and all levels of landscape fragmentation. The accuracy provided by the MPG was quite similar to the one of DIST50 although the DIST50 outperformed the MPG when landscape fragmentation was low and the foraging or foraging and territoriality processes were modelled, or when landscape fragmentation was high and the foraging process was simulated (Table 1). Note that the accuracy of both DIST50 and MPG dropped around 0.25 in highly fragmented landscape when foraging and predator avoidance or foraging and territoriality were simulated (Table 1). Case study

Identi cation of resource patches for Alpine ibex

Habitat selection by male ibex in June depended on both the distance to steep slope and land cover class (Table S4). Males selected notably low ligneous and avoided mineral and snow cover when compared to grasslands (i.e., reference habitat). Besides, they selected cover that were close to steep slope (Table S4). Ibex resource patches were highly aggregated as the aggregation index equalled 78.4 and the patch cohesion index reached 96.2. However, the division index was equal to 0.90 indicating that despite being aggregated, patches were also highly divided in the landscape. The proportion of cells occupied by habitat patches totalled 23.8%. Consequently, the level of fragmentation in ibex landscape was relatively similar to the one of landscapes de ned as medium or low fragmented in our simulations, according to these four indexes of landscape fragmentation.

Evaluation of Alpine ibex spatial network

Similarly to the simulation results, topology of ibex spatial network was impacted by sampling frequency as node degree increased signi cantly with the number of relocations used to build individual trajectory ( , Fig. 5). The amount of variance explained ( ) was however quite low indicating that other factors should impact patch connectivity more strongly. In addition, most ibex networks were scale-free according to the estimation of scale parameter (i.e., , range for scale-free networks [min-max]: 1.03-1.30, of tted models [min-max]: 0.47-0.75). However, two networks (i.e., sampling relocations every 30 seconds and 1 minute) did not show scale-free property (i.e., = 0.94, R 2 = 0.49 and = 0.93, R 2 = 0.47) even though their scale parameter were both very close to one. Furthermore, most ibex networks had the small-world property according to the comparison of their clustering coe cient and characteristic path length to the one of random graphs (Table 2). This property was however not detected when sampling frequency was too coarse (i.e., relocations every 6 hours, Table 2). A strong loss in information on animal movement appeared quickly when sampling frequency became coarser (Figs. 67). For example, when relocations were collected every 30 minutes, we detected 76% of visited patches and 52% of realized links and when relocations were collected every 2 hours, these values dropped to 51% and 26%, respectively. This loss of information did not evolve linearly with sampling frequency as a large amount of information was lost within the range of one relocation every 5 minutes to one relocation every 30 minutes (Fig. 7). Sampling frequency also impacted the detected pattern of connectivity as many identi ed links were actually spurious (Fig. 7). For example, 20% of detected links were spurious when relocations were sampled every 20 minutes and it raised to 27% when relocations were collected every hour (Fig. 7).

Commonly used theoretical networks showed relatively poor capacity to accurately predict landscape connectivity derived from our empirical dataset on Alpine ibex. Indeed, Cohen's Kappa equalled 0.29, 0.10, \varvecα α 0.26 and 0.00 for the DIST50, DIST95, MPG and random graphs, respectively. Consequently, the DIST50 predicted more accurately inter-patch movements for ibex than the MPG, even though the difference is weak, but it only predicted successfully about one-third of landscape connectivity.

Discussion

Using both simulated and empirical movement trajectories at very high-resolution of individuals showing different movement behaviours in landscapes with varying levels of fragmentation, we rst demonstrated that an accurate assessment of landscape connectivity pattern can be hampered when the relocation sampling frequency is too coarse compared to the temporal dynamics of animal movement. However, we also revealed that the level of landscape fragmentation and the processes driving animal movement can both mitigate the effect of relocation sampling frequency on the detection of accurate landscape connectivity patterns. In addition, we showed that network topologies emerging from different movement behaviours and a wide range of landscape fragmentation were generally complex, and that commonly used theoretical networks, usually of simple topologies, did not perform well to accurately predict landscape connectivity in such environments. Finally, we also pinpointed that a high amount of information is rapidly lost on landscape connectivity when sampling frequency of individual relocations becomes coarser.

Empirical spatial networks have been built from individual trajectories obtained at varying sampling frequency without knowing so far the potential effect of trajectory resolution on the assessment of landscape connectivity using networks [6, [START_REF] Wittemyer | Graph theory illustrates spatial and temporal features that structure elephant rest locations and re ect risk perception[END_REF]18,[START_REF] Prima | A landscape experiment of spatial network robustness and space-use reorganization following habitat fragmentation[END_REF]36,[START_REF] Rhodes | Applying network analysis to the conservation of habitat trees in urban environments: a case study from Brisbane, Australia[END_REF]76]. Here, we demonstrated that the sampling frequency of individual relocations can notably impede the detection of network small-world property but the strength of this effect was strongly linked to the level of landscape fragmentation. Indeed, a higher number of relocations (i.e., ner sampling frequency) was necessary to detect the small-world property of simulated networks in low fragmented than in highly or medium fragmented landscapes. The characteristics of small-world network result in many connected triplets of nodes (i.e., potential local functional use) with e cient movement within the network (i.e., through shortcuts) [56,58]. Consequently, the missed detection of such patterns could result in inaccurate understanding of how landscapes are connected and for example, could lead to missed identi cation of patches or connections used for speci c functional attributes (e.g., stopovers or stepping-stones) [START_REF] Fox | Herbivores in a small world: network theory highlights vulnerability in the function of herbivory on coral reefs[END_REF][START_REF] Saura | Stepping stones are crucial for species' long-distance dispersal and range expansion through habitat networks[END_REF][START_REF] Monteith | Functional attributes of ungulate migration: landscape features facilitate movement and access to forage[END_REF]. Similarly, we did not detect the smallworld property of Alpine ibex empirical network when sampling frequency was too coarse (i.e., one location every 6 hours). On the other hand, the scale-free property of simulated networks was always detected independently of the grain of relocation sampling frequency. However, the skewness of network degree distribution was impacted by both the level of landscape fragmentation and sampling frequency. Indeed, the local connectivity (i.e., node degree) was generally less heterogeneous (i.e., less skewed) in patchy landscapes when full trajectories were used to infer landscape connectivity but when sampling frequency became coarser, the skewness of degree distribution became stronger. These results indicate that commonly used discontinuous trajectories could lack robustness to infer landscape connectivity in realworld systems.

While habitat patch characteristics can potentially reduce the effect of sampling frequency on detected patterns of connectivity, the mitigation could yet be effective in opposite landscapes (i.e., high vs low fragmentation) depending on network topology (i.e., scale-free or small-world). Notably, in low fragmented landscapes, ne sampling frequency should be set to correctly detect small-world property while in highly fragmented landscapes, ne sampling frequency should be set to accurately assess scale-free property, according to our simulations. Many studies demonstrated that animals adjust their behaviour regarding patch selection and intensity of use as a response to patch environmental attributes [78][START_REF] Mcmahon | Habitat selection differs across hierarchical behaviors: selection of patches and intensity of patch use[END_REF][80][81][82]. We indeed demonstrated that movements in landscape having different attributes can generate different connectivity patterns and that such patterns actually require ne sampling frequency to be correctly appraised. For example, we showed that about 50% of information on ibex movements in a low to medium fragmented environment was lost when ibex were relocated every 30 minutes and about 25% of their inferred movements were spurious with this sampling rate. While vertebrate movements are generally drawn from discontinuous individual trajectories collected at a coarser rate than one location per hour [36,18,83,84,76], our empirical study rised a warning on the use of such movement inference to the assessment of landscape connectivity from spatial networks.

The effect of relocation sampling frequency on detected patterns of landscape connectivity was also largely mitigated when simulated individuals were territorials in all types of fragmented landscapes. In the simulation, territorial individuals constrained their movements to their own territory due to the presence of conspeci c scent-markings that were avoided [START_REF] Theng | Confronting spatial capture-recapture models with realistic animal movement simulations[END_REF]. Consequently, territorial individuals had access to a smaller part of the landscape than non-territorial individuals (i.e., F, F + Pe and F + Ps scenarios) such that movements within their territory were probably more redundant resulting in less information loss from coarser sampling frequency. While our simulations provide new insights on the interplay between animal movement behaviour, landscape fragmentation and relocation sampling frequency on detected patterns of landscape connectivity, it has yet to be appraised using more empirical studies to provide additional support to our results. It could notably be helpful to further improve the design of studies aiming at evaluating landscape connectivity from empirical networks.

Most of our results, either from empirical or simulated cases, conclude that very high-resolution trajectories should be used to correctly infer patterns of landscape connectivity from empirical networks. New technological developments in animal tracking should help meeting this need. Indeed, a wide range of species, from small to large size, can now be tracked using GPS tags that collect more frequent data and on longer period that in the past (e.g., by relying on solar energy) [85]. GPS relocations can be collected as frequently as every seconds, but generally at a cost of limited duration of tracking due to the high battery needs [86]. The application of dead-reckoning techniques to biologging and GPS data could also help collecting very high-resolution trajectories over longer periods [87]. DR relies on accelerometry and magnetometry data to calculate speed and heading along individual trajectory that can further be translated into very high-resolution coordinates (e.g., location every second, [67]). As battery constraints are less limited for magnetometers and accelerometers, sensor data can be collected much more frequently than solely GPS data and, when combined together to correct for magnetic drift, proved to be quite performant [67,87]. In this study, Alpine ibex were equipped with tri-axial accelerometer and tri-axial magnetometer sensors recording at 25 Hz/ 8 Hz, respectively, and we combined those data with GPS data collected every two hours to produce locations every second [67]. DR has largely been used on domestic and aquatic wild species but its application remains rare for terrestrial wild species while it could provide detailed information about the movement paths of animals between GPS xes and open new opportunities in movement ecology [88][89][START_REF] Dewhirst | Improving the accuracy of estimates of animal path and travel distance using GPS drift-corrected dead reckoning[END_REF][START_REF] Bidder | The need for speed: testing acceleration for estimating animal travel rates in terrestrial dead-reckoning systems[END_REF][START_REF] Munden | Making sense of ultrahighresolution movement data: A new algorithm for inferring sites of interest[END_REF].

Spatial networks re ecting movements among resource patches have been identi ed as having complex topologies in a large body of litterature for various species [3,[START_REF] Fox | Herbivores in a small world: network theory highlights vulnerability in the function of herbivory on coral reefs[END_REF][START_REF] Prima | A landscape experiment of spatial network robustness and space-use reorganization following habitat fragmentation[END_REF][START_REF] Rhodes | Applying network analysis to the conservation of habitat trees in urban environments: a case study from Brisbane, Australia[END_REF][START_REF] Lookingbill | Combining a dispersal model with network theory to assess habitat connectivity[END_REF]. We suggested that network topologies could be simpler than actually detected due to missed stepping-stone movements from coarse sampling frequency and consequently be more similar to simple topologies like the MPG or distancebased networks. Here, we refuted this hypothesis as all networks obtained from very high-resolution trajectories were actually complex (i.e., always scale-free and most of the time small-world). Observations of complex patterns of landscape connectivity could actually be expected as animal movement results from the complex interplay of animal state, navigation and motion capacity and external factors [START_REF] Nathan | A movement ecology paradigm for unifying organismal movement research[END_REF]. Consequently, commonly used theoretical networks did not succeed to accurately predict landscape connectivity in our simulated or empirical landscapes. These network models make general and relatively simple hypothesis on the drivers of animal movement among resource patches and are actually quite commonly used as decision-making tools [START_REF] Leblond | Science-informed policy decisions lead to the creation of a protected area for a wide-ranging species at risk[END_REF][START_REF] Bourdouxhe | Comparison of methods to model species habitat networks for decision-making in nature conservation: The case of the wildcat in southern Belgium[END_REF][START_REF] Martinez | L'aménagement urbain: (En) quête sur la préservation des espaces ouverts : L'exemple de la métropole Grenoble-Alpes[END_REF][96]. However, we demonstrated that they should be used with caution as they only succeeded to correctly predict between 30% and 50% of landscape connectivity when tested for various movement behaviours in different landscapes. Other tools such as individual based models (IBMs), have been combined with network theory to assess landscape connectivity [START_REF] Grimm | Pattern-oriented modeling of agent-based complex systems: Lessons from ecology[END_REF]98]

Conclusions

Animal behaviour, the level of landscape fragmentation and the frequency of individual relocations were shown to have an effect on the assessment of landscape connectivity using spatial networks. In general, very high-resolution trajectories (e.g., relocations collected every 1 minute in Alpine ibex) were necessary to accurately identify complex network topologies and avoid the generation of spurious information on landscape connectivity. New technologies providing such high-resolution datasets over long period, as for example dead-reckoning techniques, should thus grow in the movement ecology sphere to help unravelling many ecological questions and conservation challenges. In addition, commonly used theoretical models should be applied with caution to the study of landscape connectivity in real-world systems as they did not perform well as predictive tools. Instead, other modelling tools should be challenged to assess whether they can correctly generate more realistic functional patterns of landscape connectivity in actual environments.

General description of network topologies and design of the study. Ibex photo ©: Franck Merlier.

Degree distributions of simulated networks in three types of landscape fragmentation from four movement processes. F: Foraging, F+Pe: Foraging and avoidance of an elusive predator, F+Ps: Foraging and avoidance of a stalking predator, F+T: Foraging and territoriality. All distributions decayed as a power function with scale parameter being higher than 1 (Table S3).

Estimates and their 95% con dence interval of average node degree of simulated networks. One model was run for each simulated movement process: F: Foraging, F+Pe: Foraging and avoidance of an elusive predator, F+Ps: Foraging and avoidance of a stalking predator, F+T: Foraging and territoriality. Each model included as covariates the level of landscape fragmentation, the number of relocations used to build the spatial network (correlated to the sampling frequency) and the interaction of both. Pseudo-R 2 of the models were: F: 0.14 ; F+Pe: 0.18 ; F+Ps: 0.18 ; F+T: 0.26.

Pourcentage of simulated networks having the small-world property. The small-world property is de ned as a relatively high clustering coe cient and a similar characteristic path length compared to a random graph. The number of relocations is correlated to the sampling frequency with higher number of relocations indicating ne sampling frequency. The combination of three levels of landscape fragmentation and four movement processes were simulated: F: Foraging, F+Pe: Foraging and avoidance of an elusive predator, F+Ps: Foraging and avoidance of a stalking predator, F+T: Foraging and territoriality.

2 =

 2 log (D) = 0.043 (±0.010) * log (N ) , p < 0.001, R

Figure 6

 6 Figure 6

Table 1

 1

	Adequacy (Cohen's Kappa) between theoretical spatial networks and simulated networks from four
						movement processes.
		F			F + Pe			F + Ps			F + T
		Low	Med	High	Low	Med	High	Low	Med High	Low	Med High
	Dist50	0.46	0.42	0.40	0.49	0.42	0.24	0.48	0.41	0.24	0.43	0.42	0.26
	Dist95	0.30	0.16	0.13	0.33	0.16	0.06	0.28	0.14	0.05	0.41	0.29	0.09
	MPG	0.39	0.41	0.31	0.48	0.42	0.23	0.47	0.41	0.24	0.31	0.42	0.26
	Rand	0.01	-0.00	0.01	0.00	-0.00	0.00	0.00	0.00	-0.00	0.00	0.01	0.03
	Legend of

Table 1 :

 1

The Cohen's Kappa is calculated from the comparison of predicted links from four theoretical spatial networks and observed links obtained by simulation of individual trajectory from different movement processes in landscapes having various levels of fragmentation (low, medium (med) or high). Simulated movement processes: F: Foraging, F + Pe: Foraging and avoidance of an elusive predator, F + Ps: Foraging and avoidance of a stalking predator, F + T: Foraging and territoriality. Theoretical networks: DIST50: distance-based network where patches closer than L 50% are connected, DIST95: distance-based network where patches closer than L 95% are connected, MPG: minimum planar graph, Rand: random network where random patches are connected. L 50% is the median length of realized links obtained from unique inter-patch moves of the complete simulated trajectory and L 95% is the 95% quantile of the distribution of realized link length. Kappa can vary between - 1 and 1 with closer values to 1 indicating perfect match.

Table 2

 2 Properties of Alpine ibex spatial networks in the Belledonne massif (French Alps) in June 2017.

	Sampling	Num. of	Num. of	Num.		Scale-	C	L	Small-
	frequency	locations	visited	of links		free			world
			patches
					[R 2 ]
	Every 1sec	18 070	83	170	1.03 [0.60]	Yes	0.42	4.84	Yes
		914
	Every 30sec	604 720	80	166	0.94 [0.49]	No	0.44	4.84	Yes
	Every 1min	302 380	80	167	0.93 [0.47]	No	0.44	4.86	Yes
	Every 5min	60 480	76	154	1.03 [0.65]	Yes	0.39	4.75	Yes
	Every	30 240	73	142	1.15 [0.75]	Yes	0.46	4.79	Yes
	10min
	Every	15 120	67	133	1.15 [0.71]	Yes	0.42	4.79	Yes
	20min
	Every	10 080	63	119	1.10 [0.68]	Yes	0.43	4.54	Yes
	30min
	Every 1hour	5 040	55	100	1.10 [0.74]	Yes	0.49	3.75	Yes
	Every	2 619	42	59	1.22 [0.75]	Yes	0.22	3.22	Yes
	2hours
	Every	1 005	28	28	1.30 [0.47]	Yes	0.11	3.71	No
	6hours
	Legend of

Table 2 :

 2 Sampling frequency indicates the frequency at which we resampled the full individual

trajectories (i.e., every 1sec) to further build ibex spatial network. [R 2 ]: scale parameter estimate of network degree distribution and its associated model's R-squared, C: network clustering coe cient, L: network characteristic path length.

  . An IBM models individual movement in a landscape from parameterized rules of movements and consequently allows to generate mechanistic-based stochastic movement trajectories in heterogeneous environments[START_REF] Lookingbill | Combining a dispersal model with network theory to assess habitat connectivity[END_REF][START_REF] Da Rocha | Dispersal movement through fragmented landscapes: the role of stepping stones and perceptual range[END_REF] 100]. Many IBMs have been developped to simulate dispersal behaviour in heterogeneous landscapes (i.e., individual based dispersal models [98]) notably because IBMs can address questions even when empirical knowledge is insu cient for linking individual-level processes to landscape-level patterns[101]. However, the application of IBM to the study of landscape connectivity using networks in the context of home-ranging behaviour (e.g., including recursive movements [102, 103]) remains underexploited and could cope with the urgent need to develop theoretical network models that t with complex but more realistic functional patterns of connectivity[101].

Acknowledgements: The authors thank all participants for helping with the capture and marking of ibex and for the recovery of GPS collars: volunteers and professionals from the O ce Français de la Biodiversité (Carole Toïgo, Mathieu Beurier and SD38).

Availability of data and materials: The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

Supplementary Files

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SupportingInformation.docx


Denis V Anokhin

Ludmila L Gur'eva
email: gurieva@icp.ac.ru

Kseniia N Grafskaia

Evgeniy S Pikalov

Ainur F Abukaev
email: abukaev.af@phystech.edu

Viktor P Tarasov
email: tarasov.07@list.ru

Dimitri A Ivanov
email: dimitri.ivanov@uha.fr

Control of self-assembly of amphiphilic wedge-shaped mesogens by combined magnetic field and temperature treatment

Keywords: wedge-shaped amphiphilic mesogens, 2,3,4-tris-(dodecyloxy)benzene sulfonates, 36 orientation in magnetic field, differential scanning calorimetry, grazing-incidence X-ray diffraction

The report elucidates for the first time a significant effect of strong magnetic field combined with thermal treatment on the texture of thin liquid-crystalline films in a smectic state. The metastable texture generated in magnetic field was arrested by crystallization of mesogens during cooling to room temperature. The effect was demonstrated on a series of wedge-shaped amphiphilic mesogens based on 1,2,3-tris-(dodecyloxy)benzene (TDOB): asymmetric 2,3,4-tris-(dodecyloxy)benzene sulfonic acid (TDOBSH) and it's sodium (TDOBSNa) and pyridine (TDOBSPyr) salts. The thermotropic properties and structure of liquid crystal phases of synthesized compounds were addressed by differential scanning calorimetry, polarized optical microscopy and X-ray diffraction. It was shown that depending on the type of counterion the synthesized 28 mesogens form different supramolecular structures. The largest effect of intense mag-29 netic field was observed for the pyridine salt for which an ordered primitive cubic phase 30 texture was generated upon applying magnetic field normal with respect to the film. In 31 contrast, for the corresponding acid an improvement of gyroid cubic phase orientation 32 was detected under magnetic field oriented along the film. A highly ordered columnar 33 phase of the sodium salt showed a minimal effect of magnetic field. 34 35

Introduction

One of the most powerful tools employed for development of smart, sensitive ma-40 terials is the molecular self-assembly [START_REF] Antipin | Functional supramolecular systems: design and applications[END_REF]. The unique properties of supramolecular such molecular "building blocks" is determined both by the shape and size of the original molecules and the position of functional groups [START_REF] Alfutimie | Lyotropic Surfactant Liquid Crystals: Micellar Systems[END_REF]. External factors shift the balance of these interactions stimulating a change in molecular conformation and, consequently, phase transition to a new structure. In particular, application of magnetic field could in some instances generate a texture which would be not thermodynamically stable and therefore not easily achievable in the absence of the field such as the so-called homeotropic orientation discussed later.

Wedge-shaped amphiphilic mesogens based on tris(alkyloxy)benzene sulphonic acid is a relatively new class of self-assembling molecules [START_REF] Li | Light-Switchable Vesicles from Liquid-Crystalline Homopolymer-Surfactant Complexes[END_REF][START_REF] Zhang | From Channel-Forming Ionic Liquid Crystals Exhibiting Humidity-Induced Phase Transitions to Nanostructured Ion-Conducting Polymer Membranes[END_REF]. They can serve as building blocks for synthesis of novel functional materials, such as highly regular polymeric ion-conductive membranes, nanoreactors and drug delivery systems. Variation of chemical composition (change in alkyl chain length, substitution of counterion, addition of azobenzene group) or external factors (temperature change, presence of saturated solvent vapor atmosphere with different polarity, UV-irradiation) allow to form crystalline or liquid-crystalline phases of different symmetry.

Self-assembly of amphiphilic mesogens seems as a promising "bottom-up" approach for fabrication of membranes with highly-defined size and topology of ion channels which can become alternative to Nafion or zeolites as efficient proton-conductive membrane for application of perspective fuel cells. [START_REF] Mauritz | State of Understanding of Nafion[END_REF][START_REF] Allison | Sorption in the β-phases of transition metal(II) tetra-(4-methylpyridine) thiocyanates and related compounds[END_REF][START_REF] Barrer | Ion-Exchanged Forms of Zeolite L, Erionite, and Offretite and Sorption of Inert Gases[END_REF]. The proton transport across the membranes built from such molecules can be optimized using external stimuli during film preparation [START_REF] Grafskaia | on-off" switchable cubic phase with exceptional thermal stability and water sorption capacity[END_REF]. The most common mesophase structures for thin films of mesogens containing ion channels are columnar (hexagonal and monoclinic) and smectic. Such liquid-crystalline films demonstrate textures having mainly in-plane orientation of the channels which is not optimal for proton transport across the membrane. To achieve optimal membrane efficiency, it is necessary to increase the number of channels for proton transport across the membrane, which is possible when the channels are normally, or homeotropically, oriented with respect to the membrane film surface. However, switching the channels' orientation to normal or homeotropic is limited by thermodynamic instability of such a texture [START_REF] Sirringhaus | Two-dimensional charge transport in self-organized, high-mobility conjugated polymers[END_REF][START_REF] Hosono | Large-Area Three-Dimensional Molecular Ordering of a Polymer Brush by One-Step Processing[END_REF][START_REF] Grelet | Control of the orientation of thin open supported columnar liquid crystal films by the kinetics of growth[END_REF]. One of the ways to provide stable homeotropic texture in thin films is to apply a magnetic field to the mesogens in isotropic or LC state and then fix it by chemical bonding [START_REF] Majewski | Anisotropic Ionic Conductivity in Block Copolymer Membranes by Magnetic Field Alignment[END_REF][START_REF] Tousley | Aligned Nanostructured Polymers by Magnetic-Field-Directed Self-Assembly of a Polymerizable Lyotropic Mesophase[END_REF][START_REF] Vita | Molecular ordering in the high-temperature nematic phase of an all-aromatic liquid crystal[END_REF][START_REF] Feng | Scalable Fabrication of Polymer Membranes with Vertically Aligned 1 nm Pores by Magnetic Field Directed Self-Assembly[END_REF]. The membranes based on polymerizable wedge-shaped amphiphilic LCs were shown to form highly ordered mesophases in magnetic field resulting an increase of their conductivity by two orders of magnitude as compared to unoriented materials [START_REF] Feng | Scalable Fabrication of Polymer Membranes with Vertically Aligned 1 nm Pores by Magnetic Field Directed Self-Assembly[END_REF]. It is important to mention that reorientation of mesogens in magnetic field mainly occurs in nematic LC state which is characterized by low degree of molecular ordering. For columnar phases, magnetic field was found to only enhance the thermodynamically stable planar orientation [START_REF] Feng | Scalable Fabrication of Polymer Membranes with Vertically Aligned 1 nm Pores by Magnetic Field Directed Self-Assembly[END_REF]. [START_REF] Zhang | From Channel-Forming Ionic Liquid Crystals Exhibiting Humidity-Induced Phase Transitions to Nanostructured Ion-Conducting Polymer Membranes[END_REF][START_REF] Zhu | Wedge-Shaped Molecules with a Sulfonate Group at the Tip-A New Class of Self-Assembling Amphiphiles[END_REF][START_REF] Pifferi | New compounds: Synthesis of 3,4,5-trimethoxybenzenesulfonamides[END_REF][START_REF] Zhu | Self-Organization of Polybases Neutralized with Mesogenic Wedge-Shaped Sulfonic Acid Molecules: An Approach toward Supramolecular Cylinders[END_REF] and asymmetric [START_REF] Li | Light-Switchable Vesicles from Liquid-Crystalline Homopolymer-Surfactant Complexes[END_REF][START_REF] Zhu | Methacrylated Self-Organizing 2,3,4-Tris(alkoxy)benzenesulfonate: A New Concept Toward Ion-Selective Membranes[END_REF][START_REF] Beginn | Thermotropic columnar mesophases of wedge-shaped benzenesulfonic acid mesogens[END_REF][START_REF] Shcherbina | The effect of the shape of the mesogenic group on the structure and phase behavior of 2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline cations[END_REF][START_REF] Shcherbina | Heuristics for precise supramolecular control of soft matter structure and properties -2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline and organic cations[END_REF] tris(alkyloxy)benzene sulfonic acid exhibit a variety of both crystalline and thermotropic liquid-crystalline mesophases, including smectic, cubic, ordered and disordered columnar mesophases [START_REF] Zhu | Methacrylated Self-Organizing 2,3,4-Tris(alkoxy)benzenesulfonate: A New Concept Toward Ion-Selective Membranes[END_REF][START_REF] Beginn | Thermotropic columnar mesophases of wedge-shaped benzenesulfonic acid mesogens[END_REF][START_REF] Shcherbina | The effect of the shape of the mesogenic group on the structure and phase behavior of 2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline cations[END_REF][START_REF] Shcherbina | Heuristics for precise supramolecular control of soft matter structure and properties -2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline and organic cations[END_REF]. These phases can be stabilized by physical and chemical cross-linking to form polymer membranes with well-controlled ion channel topologies [START_REF] Zhang | From Channel-Forming Ionic Liquid Crystals Exhibiting Humidity-Induced Phase Transitions to Nanostructured Ion-Conducting Polymer Membranes[END_REF][START_REF] Grafskaia | on-off" switchable cubic phase with exceptional thermal stability and water sorption capacity[END_REF].

Derivatives of symmetric

Summarizing, the development of approaches to control the orientation of ion-conducting channels in self-organized systems of wedge-shaped amphiphilic LC compounds is of great interest [START_REF] Li | Light-Switchable Vesicles from Liquid-Crystalline Homopolymer-Surfactant Complexes[END_REF][START_REF] Zhang | From Channel-Forming Ionic Liquid Crystals Exhibiting Humidity-Induced Phase Transitions to Nanostructured Ion-Conducting Polymer Membranes[END_REF][START_REF] Grafskaia | on-off" switchable cubic phase with exceptional thermal stability and water sorption capacity[END_REF][START_REF] Zhu | Wedge-Shaped Molecules with a Sulfonate Group at the Tip-A New Class of Self-Assembling Amphiphiles[END_REF][START_REF] Pifferi | New compounds: Synthesis of 3,4,5-trimethoxybenzenesulfonamides[END_REF][START_REF] Zhu | Self-Organization of Polybases Neutralized with Mesogenic Wedge-Shaped Sulfonic Acid Molecules: An Approach toward Supramolecular Cylinders[END_REF][START_REF] Zhu | Methacrylated Self-Organizing 2,3,4-Tris(alkoxy)benzenesulfonate: A New Concept Toward Ion-Selective Membranes[END_REF][START_REF] Beginn | Thermotropic columnar mesophases of wedge-shaped benzenesulfonic acid mesogens[END_REF][START_REF] Shcherbina | The effect of the shape of the mesogenic group on the structure and phase behavior of 2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline cations[END_REF][START_REF] Shcherbina | Heuristics for precise supramolecular control of soft matter structure and properties -2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline and organic cations[END_REF]. In the present work, a combined effect of high-temperature annealing and magnetic field was addressed for wedge-shaped amphiphilic mesogens: asymmetric 2,3,4-tris-(dodecyloxy)benzene sulfonic acid (TDBОSН) as well as its sodium (TDBОSNa) and pyridine (TDBОSPyr) salts. These compounds demonstrate complex thermotropic behavior with formation of both crystalline and liquid crystalline phases. This opens perspectives for design of thin films with unusual texture by application of strong external stimulus like magnetic field on the films in LC state with high molecular mobility with following fixing of desired supramolecular morphology by crystallization at room temperature.

Materials and Methods

Synthesis

The TDBОSН, TDBОSNa and TDBОSPyr were synthesized according to the following scheme by the methods described in [START_REF] Zhang | From Channel-Forming Ionic Liquid Crystals Exhibiting Humidity-Induced Phase Transitions to Nanostructured Ion-Conducting Polymer Membranes[END_REF][START_REF] Zhu | Methacrylated Self-Organizing 2,3,4-Tris(alkoxy)benzenesulfonate: A New Concept Toward Ion-Selective Membranes[END_REF][START_REF] Beginn | Thermotropic columnar mesophases of wedge-shaped benzenesulfonic acid mesogens[END_REF][START_REF] Shcherbina | The effect of the shape of the mesogenic group on the structure and phase behavior of 2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline cations[END_REF][START_REF] Shcherbina | Heuristics for precise supramolecular control of soft matter structure and properties -2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline and organic cations[END_REF]: Scheme 1. Synthesis of TDOB, TDBОSН, TDBОSNa and TDBОSPyr.

Reagents

Pyrogallol (99%), 1-bromodododecane (95%), pyridine (99.8%), sodium methylate (pure, anhydrous) (Sigma-Aldrich), benzene (Cd, Ecos), acetone, sulphuric acid, potassium carbonate (CH, Chemmed) were used without further purification. Molecular sieves 4A (granules, diameter 3.2 mm, Acros Organics) were calcinated under vacuum at 250 oC. Dimethylformamide (DMFA) (99.8%, Sigma-Aldrich) was dehydrated by azeotropic distillation with benzene at atmospheric pressure, incubated over molecular sieves 4A for 3 days, followed by vacuum distillation under argon current. The methanol (Cr, Chemmed) for recrystallization was distilled over molecular sieves 4A.

Methods of the analysis

Identification of the synthesized compounds was carried out by IR, 1 H NMR and 13 C NMR spectroscopy and elemental analysis. IR spectra were taken in KBr tablets using Specord M80 spectrometer with Soft Spectra software. Two-dimensional Cosy and HSQC NMR spectra were measured in CDCl3 solution with trimethylsilane (TMS) reporter on AVANCE-III-500 spectrometer and were interpreted using Mest ReNova 12-0-0 Manual software. Elemental analysis was performed on Vario EL cube equipment (Elementar GmbH).

Thermotropic properties of compounds were determined by the differential scanning calorimetry (DSC) methods using "DSC 30 Mettler Toledo Star System" in the temperature range from -40 to 140°C with scanning rate of 5 deg/min. In order to check the reversibility of phase transitions two DSC heating-cooling ramps were performed for all samples.

Thin films (200-500 nm) for POM and GIXD experiments were prepared by spin-coating of 10 mg/ml solution in chloroform on a silicon substrate at a rate of 1000 rpm. For NMR experiments the samples were prepared by drop-casting of 20 mg/ml so-lution in chloroform on a silicon substrate. Then the drop-casted films were annealed at 85°C for 1 hour in vacuum oven.

The effect of thermal history on texture of the thin films of the synthesized compounds was studied by polarized optical microscopy (POM). The spin-coated films were annealed in normal atmosphere at 85°C for 3 hours using Linkam LTS420 thermal stage.

Optical measurements of the as-cast and annealed films were carried out with a Karl Zeiss Axio Scope A1Pol microscope in polarized light.

The structure of unoriented samples was studied by X-ray diffraction (XRD) on custom-built WAXS/SAXS X-ray System diffractometer with a 1.54 Å wavelength equipped with a two-dimensional Pilatus 300k detector. The sample-detector distance was 1.15 m. The modulus of the scattering vector s (|s| = 2sin /, where  is the Bragg angle,  is the wavelength and |s| is the norm of the s vector) was calibrated using several diffraction orders of silver behenate. To exclude the influence of humidity the measurements were carried out in vacuum. Analysis of X-ray data, including background subtraction and radial integration of 2D diffractograms was performed with procedures developed in the IgorPro (Wavemetrics Ltd.) environment.

To address the effect of magnetic field (MF) on thin film texture, the capillary with sample on substrate was placed into resonator of BRUKER-500 NMR spectrometer (proton frequency 400 MHz). Magnetic induction 9.5 Tesla was oriented normal to the capillary axis. As it was shown in [START_REF] Feng | Scalable Fabrication of Polymer Membranes with Vertically Aligned 1 nm Pores by Magnetic Field Directed Self-Assembly[END_REF] such intensity is sufficient for reorganization of benzene-containing molecules in LC phase. Orientation of B vector of MF to set it normal (Fig. 1a) or parallel (Fig. 1b) to the film surface was achieved by rotation of the capillary around its axis. Then, the samples were heated to temperatures above isotropization point (85°C for TDBОSН and TDBОSPyr, 110°C for TDBОSNa) and slowly cooled down to room temperature. During cooling NMR spectra of the films were recorded. The texture of the films after MF exposure was addressed by grazing-incidence X-ray diffraction (GIXD) at the BM26 beamline of the European Synchrotron Radiation Facility (Grenoble, France). The measurements were performed at a wavelength of 1.03 Å and the diffraction signal was recorded with a 2D Pilatus 300k detector. The indexation of 2D diffractograms was made in procedures built in IgorPro (Wavemetrics Ltd.) software.

For films oriented parallel to the MF, X-ray beam was oriented normally to the direction of MF (Fig. 1c).

Results

The DSC thermograms of the synthesized mesogens TDOBSН, TDOBSNa and TDOBSPyr are presented in Fig. 2, and their thermal parameters are summarized in Table 1. The precursor TDOB has a single endothermic melting peak of isotropization of crystalline phase (Table 1), which indicates the absence of LC state. The second endothermic peak both on first and on second DSC heating ramps in the region 90-130°С

shows isotropization of LC phase (see Figure 2 and Table 1). The found values transition temperatures are close to the literature data [3, 4, 16 -22]. To improve the texture of thin films of the studied sulphonates the samples were annealed in the temperature window of existence of LC phase between T1 and T2. From one hand, at this temperature the molecular mobility is enhanced to reach thermodynamically equilibrium phase and large ordered domains. From another hand, the thin films in LC state keep a certain mechanical stability, whereas isotropization can lead to dewetting and other negative factors. As-cast and annealed films were studied by POM.

The formation of spherulitic textures typical for crystalline material is observed for TDOBSH as-cast film (Fig. 3a). After annealing of the films, the birefringence largely disappears indicating either isotropization or formation of optically inactive cubic phase (Fig. 3b). The POM image of as-cast film of sodium salt shows optically active needle-like domains typical for smectic or columnar LC phase (Fig. 3c). Annealing at 85°C results in growth of optical activity and appearing of granular texture (Fig. 3d). Presence of bulky pyridinium counterion significantly changes self-assembly of thin film. Microphotographs of the as-cast film of TDOBSPyr show a fine-grained structure with low optical activity (Fig. 3e). Annealing leads to a decrease in optical density indicating the formation of a cubic phase (Fig. 3f). Thus, depending on the type of counterion and thermal prehistory the synthesized mesogens reveal a variety of supramolecular structures. The film was oriented horizontally. The black regions appear after background substruction.

A detailed analysis of the crystalline and LC structure of the thin films was made by GIXD technique. Investigation of the as-cast thin films of TDOBSH shows formation of smectic structure with lattice parameter asm=36.8 Å and a small fraction of gyroid cubic phase with parameter agyr=98.0 Å (Fig. 4a) [START_REF] Garstecki | Scattering Patterns of Multiply Continuous Cubic Phases in Block Copolymers. I. The Model[END_REF]. During annealing of the sample at 85°C the smectic phase transforms into thermodynamically stable gyroid structure with decreased parameter agyr=98.0 Å (Fig. 4b). The GIXD patterns of TDOBSNa indicate presence of hexagonal columnar phase with ahex = 36.6 Å in both as-cast and annealed films (Fig. 4c,d). The increase of half-width of the reflections after annealing indicates a certain growth of grain-like domains without any preferential orientation. The analysis of GIXD diffractograms of as-cast films of TDOBSPyr reveals the self-assembly similar to TDOBSH (Fig. 4e,f). During annealing the smectic phase (asm=41.3 Å) transforms to Im3hm cubic phase with ap = 48.4 Å. Formation of primitive cubic phase was detected earlier for wedge-shaped sulfonates with potassium cation at high temperature [START_REF] Shcherbina | The effect of the shape of the mesogenic group on the structure and phase behavior of 2,3,4-tris(dodecyloxy)benzenesulfonates with alkaline cations[END_REF].

Consequently, formation of Im3hm cubic phase in thin film of TDOBSPyr can be explained by the presence of bulky pyridinium counterion. The results of indexation are summarized in Table 2. One can see that variation of the type of counterion allows changing thermal and phase behavior of the synthesized mesogens, as well as the topology of ion channels. The existence of liquid-crystalline state at elevated temperature can be explained by disordering of linear alkyl chains and increase of local molecular mobility [START_REF] Grafskaia | on-off" switchable cubic phase with exceptional thermal stability and water sorption capacity[END_REF]. However, for practical application of synthesized compounds as proton-conductive membrane it is important to provide homeotropic orientation of water channels in columnar and smectic phase. After formation of such thermodynamically metastable state at elevated temperature, it can be consequently fixed by crystallization of the mesogens during cooling to room temperature without disturbance of supramolecular organization. The effective external stimulus for formation of metastable homeotropic LC texture is application of strong magnetic field assessable in NMR spectrometer.

Fig. 5 shows NMR spectra of synthesized mesogens in LC state obtained at different temperatures on a BRUKER 500 spectrometer with a proton frequency of 500 MHz.

The narrow line at ~1 ppm on NMR spectra of TDOBSH corresponds to acid protons (Fig. 5a). Similar peak is visible for TDOBSNa indicating formation of a crystalline hydrate with one water molecule (Fig. 5b). The variation of structure was detected only after cooling to 24°С. The narrow peak in Fig. 5c corresponds to the hydrogen bonding of the mesogen and pyridinium proton of TDOBSHPyr. As expected, intensity of this line increases during heating indicating the self-assembly of the material. The effect of magnetic on thin films texture was studied by GIXD using synchrotron source. Fig. 6a-c shows 2D GIXD diffractograms for TDOBSH thin film before and after MF. Application of MF parallel to the film surface results in orientation of initially isotropic gyroid cubic phase and appearance of a small fraction of smectic phase (Fig. 6b).

For the gyroid lattice the 211 vector is oriented normal to the substrate. For normal orientation of MF the GIXD pattern demonstrates two weak meridional reflections of the gyroid phase (Fig. 6c). Consequently, parallel alignment of MF stimulates development of large well-oriented domains of the cubic phase. For sodium salt we do not observe significant changes in hexagonal phase texture after application of MF parallel to the film compare to non-exposed annealed sample (Fig. 6d,e). However, normal orientation of MF causes a certain disorientation of the hexagonal phase and appearance of a weak peak in small-angle region which can be attributed to less stable smectic phase (Fig. 6f). Thus, normal orientation of MF prevents formation of large regular domains of the columnar hexagonal mesophase. For pyridinium salt self-assembly of mesogens in MF parallel to the film leads to a partial transformation of initial oriented primitive cubic phase to smectic phase (Fig. 6g,h). In contrast, MF oriented normal to the film stimulates formation of mixture of unoriented and well-oriented primitive Im3hm cubic phase with 100 vector oriented normal to the surface (Fig. 6i). The difference in the effect of MF orientation on the texture improvement for TDOBSPyr can be explained by interaction of MF with pyridinium rings. The presented diffractograms confirm the effectiveness of the magnetic field on the texture and degree of ordering of amphiphilic mesogens. The largest effect of the magnetic field is observed for the pyridine salt, which is probably explained by a weak ordering of the mesogen molecules in the initial film and magnetic susceptibility of the pyridinium. A fraction of highly ordered primitive cubic phase is formed as a result of MF exposure. For the acid, on the other hand, a decrease in the initial cubic phase is observed, possibly due to rapid cooling of the sample in the NMR instrument. For the highly ordered columnar phase of the sodium salt the effect of MF application in both directions is minimal.

Conclusions

In this work, reorganization of texture of thin liquid-crystalline films in a smectic state was shown for the first time. It is noteworthy that the field-induced texture is metastable without magnetic field (MF) but can be fixed due to ability of the compounds to crystallize at room temperature and thereby create a physical network of crystalline domains. To this end, we synthesized derivatives of a wedge-shaped amphiphilic mesogen -asymmetric 2,3,4-tris(dodecyloxy)benzene sulfonate and its pyridine and sodium salts. The processes of self-assembly of synthesized mesogens in bulk and in thin

Figure 1 .

 1 Figure 1. Schematic of the sample orientation in the capillary normal (a) and parallel (b) to the MF (pink arrow); (c) Orientation of X-ray beam in respect to MF direction during GIXD measurements.

Figure 2 .

 2 Figure 2. DSC heating (1) and cooling (2) thermograms for mesogens: (a) TDOBSН, (b) TDOBSNa (1st cycle -black line, 2nd cycle -red line), (c) TDOBSPyr. Exo peaks are positive. Scanning rate 5°C/min.

Figure 3 .

 3 Figure 3. Optical micrographs of films before and after annealing for: TDOBSH (a, b); TDOBSNa (c, d); TDOBSPyr (e, f). The scalebar is 50 mkm.

Figure 4 .

 4 Figure 4. 2D GIXD patterns of TDOBSH (a,b), TDOBSNa (c,d) and TDOBSPyr (e,f) before (a,c,e) and after (b,d,f) annealing. The most intense reflections are marked with white arrows and indexed according to smectic (sm), gyroid (gyr), columnar hexagonal (hex) and cubic primitive (p) phases.

Figure 5 .

 5 Figure 5. NMR spectra of synthesized mesogens: (а)-TDOBSH at 80°С (1), 50°С (2) and 30°С (3); (b) -TDOBSNa at 80°С (1), 24°С (2); (c) -TDOBSPyr at 80°С (1), 50°С (2), 30°С (3).

Figure 6 .

 6 Figure 6. 2D GIXD patterns of TDOBSH (a-c), TDOBSNa (d-f) and TDOBSPyr (g-i): left columnbefore application of MF, middle column -with MF parallel to the film, right column -with MF normal to the film. The most intense reflections are marked with white arrows and indexed according to smectic (sm), gyroid (gyr), columnar hexagonal (hex) and cubic primitive (p) phases. The film was oriented horizontally. Red arrow indicates direction of magnetic field in respect to the film surface. The black regions appear after background substruction.

  films were preliminarily addressed by DSC, POM and GIXD. The influence of MF oriented along or normal to the film surface on the supramolecular structure formation at different temperatures was studied. It was found that depending on the counterion nature the samples demonstrate specific thermotropic behavior and form different supramolecular structures. For thin films of TDOBSH high-temperature annealing results in transition from a metastable smectic LC phase to a cubic bicontinuous gyroid phase.Applying of MF parallel to the film surface stimulates alignment of the cubic domains. In contrast, orientation of MF normal to the substrate hinders growth of regular domains of gyroid phase during slow cooling from the LC state to room temperature. The sodium salt (TDOBSNa) demonstrates formation of columnar hexagonal LC phase irrespective of the thermal history. We speculate that application of MF along the normal direction slightly disturbs the growth of large columnar domains due to appearance of metastable smectic phase. Among all the studied molecules the most pronounced effect of MF on self-assembly was detected for the pyridinium salt. The stable Im3hm cubic phase forming during annealing completely disappears under the action of MF oriented along the TDOBSPyr film.Realignment of the mesogens results in appearance of the smectic phase typical of the as-cast films. In contrast, the normally oriented MF improves the development of large highly-oriented domains of Im3hm cubic phase during cooling. The reason of specific behavior of TDOBSPyr in MF could be due to anisotropy of magnetic susceptibility of pyridine. The obtained results can help for development of strategy for fabrication of novel proton-conductive membranes built by "bottom-up" approach. Author Contributions: Conceptualization, D.V.A.; Data curation, A.F.A.; Formal analysis, K.N.G.; Investigation, E.S.P.; Methodology, V.P.T.; Supervision, D.V.A.; writing -original draft preparation, L.L.G.; Writing -review and editing, D.A.I. All authors have read and agreed to the published version of the manuscript.

Table 1 .

 1 Thermal parameters of TDOB, TDOBSН, TDOBSNa and TDOBSPyr measured by DSC

	Sample	Т1, °С	Heating ∆Н, J/g Т2, °С	∆Н, J/g	Cooling Т1, °С Т2, °С
	TDOB	42.9	188.8	-	-	21.5	-
	TDOBSН
	1 st heating	72.7	99.0	90.0	5.1	-	4.1
	2 nd heating	6.8	-	17.0	-	10.8	-0.7
	TDOBSNa
	1 st heating	49.7	46.7	105.0	4.2	73.7	-4.8
	2 nd heating	-0.7	-	94.0	-	73.6	-4.6
	TDOBSPyr
	1 st heating	79.7	105.9	128.3	1.7	120.0	17.9

Table 2 .

 2 Indexation of GIXD patterns of thin films

	Sample	Phase	As-cast films hkl	dexp, Å	After annealing Phase hkl	dexp, Å
	TDOBSН	Sm asm = 36.9 Å Cubgyr agyr = 98.0 Å	100 200 211 310 222	36.9 18.6 41.4 29.9 28.2	Cubgyr agyr = 93.2 Å	211 220 310	37.9 33.3 29.3
	TDOBSPyr	Sm asm = 41.3 Å Cubp	100 200 300 200	41.3 20.7 13.9 24.2	Sm asm = 40.0 Å	100 200 300 100	40.0 20.0 13.3 48.4
		ap = 48.4 Å			Cubp	110	33.9
						ap = 48.5 Å	111	29.0
							200	24.2
	TDOBSNa	Colh ahex = 36.6 Å	100 110 200	31.7 18.4 15.9	Colh ahex = 36.2 Å	100 110 200	31.5 18.3 15.6

Acknowledgments:

The authors thank the IPCP RAS Analytical Center for the 1H and 13C NMR tests and for the elemental analysis and M. Rosenthal (BM26, ESRF) for GIXD experiments.

Funding: The synthetic part of the work was funded by State orders No. 0089-2019-0012 (state registration number AAA-A19-119032690060-9) and No. 0089-2019-0002 (state registration number AAA-A19-119071190017-7). The structural characterization of the phases and orientation was funded by the Sirius University of Science and Technology.

Data Availability Statement:

The data presented in this study are available in article

Conflicts of Interest:

The authors declare no conflict of interest.