Hadrien Grasland
email: grasland@lal.in2p3.fr

François Févotte

Bruno Lathuilière

David Chamont

Floating-point profiling of ACTS using Verrou

Floating-point computations play a central role in scientific computing. Achieving high numerical stability in these computations affects not just correctness, but also computing efficiency, by accelerating the convergence of iterative methods and expanding the available choices of precision. The ACTS project aims at establishing an experiment-agnostic track reconstruction toolkit. It originates from the ATLAS Run2 tracking software and has already received strong adoption by FCC-hh. It is also being evaluated for possible use by the CLICdp and Belle 2 experiments. In this study, Verrou, a Valgrind-based tool for dynamic instrumentation of floating-point computations, was applied to the ACTS codebase for the dual purpose of evaluating its numerical stability and investigating possible avenues for use of reduced-precision arithmetic.

Introduction

Floating-point computations in physics

In spite of being the closest programmatic equivalent of manual approximate computation, floating-point (FP) arithmetic is widely regarded as a complex topic [START_REF] Goldberg | What every computer scientist should know about floating-point arithmetic[END_REF]. As a result, scientists who are not numerical computing experts tend to use it in imperfect ways, for instance by assuming that FP numbers are equivalent to real numbers, or always using the largest precision that is broadly implemented in hardware, IEEE-754's "double precision" 1 . Double-precision computation, however, is neither necessary nor sufficient as an approximation of exact computation. It can be insufficient in the presence of numerically unstable algorithms which expose its inexact nature, for example through accumulation of many small numbers in an indefinitely growing accumulator, or subtraction of two numbers of similar magnitude. And it can be unnecessary in the presence of input data which is far less precise than its relative accuracy (around 10 -16 ), and handled by sufficiently stable algorithms.

On modern computer hardware, and in the face of the computational challenges of future HEP experiments, there is a strong incentive to prefer using single-precision IEEE-754 arithmetic, which has broader hardware support, uses twice less memory resources (bandwidth, caches...), reduces the need for many internal iterations in IEEE-754-compliant transcendental functions, and enables wider vectorization. However, determining where this precision is applicable can be challenging, especially in large codebases which were not designed for it.

As we shall see, dynamic program instrumentation can help address these challenges.

ACTS (A Common Tracking Software)

ACTS [START_REF] Gumpert | ACTS [software[END_REF] is a free and open-source software project for track reconstruction in high-energy physics (HEP) experiments. As a modernized version of the particle tracking code used by the ATLAS experiment [START_REF]Technical Design Report for the ATLAS Inner Tracker Strip Detector[END_REF][START_REF]Technical Design Report for the ATLAS Inner Tracker Pixel Detector[END_REF][START_REF]ATLAS experiment at CERN[END_REF] during Run 2 of the Large Hadron Collider, the project is focused on adoption of modern C++ standards, usability in multi-threaded workflows, and extended use of vectorization. Key features include:

• Constructing a tracking geometry description from TGeo, DD4Hep, or GDML input

• A simple and efficient event data model

• Performant and highly flexible algorithms for track propagation and fitting

• Basic seed finding algorithms

A key aim goal of the project is to support the ever increasing needs of future accelerators such as HL-LHC [START_REF]The HL-LHC project[END_REF] and FCC [START_REF]Future Circular Collider study[END_REF].

Verrou: A dynamic floating-point instrumentation tool

Verrou [START_REF] Févotte | Verrou [software[END_REF][START_REF] Févotte | Verrou: Assessing floating-point accuracy without recompiling[END_REF] is a tool aiming to help diagnose, debug and optimize FP computations in large, industrial scientific computing codes. As an example, Verrou has already been used to debug code_aster, a structural mechanics simulation tool of more than 1.2M lines of code [START_REF] Févotte | Studying the Numerical Quality of an Industrial Computing Code: A Case Study on code_aster[END_REF].

From a user perspective, Verrou performs Dynamic Binary Instrumentation (DBI) on the analyzed program, replacing each FP instruction with a variant implementing another type of arithmetic. Program execution is otherwise left unperturbed, resulting in output which is similar to that of a standard execution. Analyzing the observed changes in the output allows estimating the global impact of FP arithmetic during program execution.

Verrou performs this instrumentation by building on top of Valgrind [START_REF] Nethercote | Valgrind: A Framework for Heavyweight Dynamic Binary Instrumentation[END_REF]. This enables high usability, as there is no need to manually change the program or recompile it; all that is needed is to prefix the usual command with an invocation of Valgrind with the Verrou tool:

valgrind --tool=verrou [VERROU_ARGS] PROGRAM [ARGS] It is worth noting that DBI in general (and Valgrind in particular) naturally composes well with other tools which might be used in the analyzed program: compilers and specific compilation options, (potentially closed-source) third-party libraries, parallelization frameworks. . .

Floating-Point Arithmetic variants

In the command above, [VERROU_ARGS] allows changing several aspects in the behavior of Verrou, and most notably the type of alternate FP arithmetic which is to be introduced in the analyzed program. Verrou implements a few variants of FP arithmetic:

Fixed rounding mode: in this mode, the rounding mode can be fixed to any of the 4 standard rounding modes defined by IEEE-754: rounding upwards, downwards, toward zero or to the nearest FP number. Additionally, a 5 th non-standard rounding mode can be emulated, which always rounds an FP calculation in the opposite way to the standard rounding to nearest. As discussed in [START_REF] Kahan | How futile are mindless assessments of roundoff in floating-point computation[END_REF], some insight on FP errors can be obtained from the comparison of a few results obtained with the same program, using different rounding modes.

Stochastic rounding mode: in this mode, the result of every FP instruction is randomly rounded upwards or downwards. Depending on the chosen probability law, this can be similar to an asynchronous CESTAC arithmetic [START_REF] Vignes | A stochastic arithmetic for reliable scientific computation[END_REF] or a variant of Monte-Carlo arithmetic [START_REF] Parker | Monte Carlo arithmetic: exploiting randomness in floating-point arithmetic[END_REF]. Numerous works in the literature detail the statistical post-processing techniques which can be used to assess the impact of FP arithmetic by analyzing the results of several randomly rounded executions of the program [START_REF] Parker | Monte Carlo arithmetic: exploiting randomness in floating-point arithmetic[END_REF][START_REF] Chesneaux | On the robustness of the CESTAC method[END_REF][START_REF] Sohier | Confidence intervals for stochastic arithmetic[END_REF].

Single precision: in this mode, the result of every double-precision operation is rounded as if it had had been performed in single precision. This feature can be an easy way to simulate a single-precision version of any program, without having to change its source code.

Instrumentation scope

Algorithms which are not robust to changes in the FP rounding modes often indicate issues in the analyzed program: for example, such an algorithm will produce varying results if parallelization and/or vectorization changes its execution order. However, some algorithms rightfully demand that a standard rounding be used. For example, the trignonometric functions of the standard mathematical library (libm), require rounding to the nearest result.

In order to handle such cases, Verrou features the ability to restrict the scope of the instrumentation: it can be instructed to leave some parts of the program unperturbed. This can be performed at the granularity of functions (or, rather symbols in the object files and libraries composing the instrumented executable binary), but also at the granularity of source code lines (if debugging information are available2 ).

Like Valgrind, Verrou only instruments the program under study by default. But in multiprocess applications, like some test runners, instrumenting child processes can be required. This can be done by adding the --trace-children=yes flag to the command above.

Debugging

While simply changing the underlying arithmetic can help assessing and diagnosing FPrelated issues in a given program, more is needed in order to help debugging them. The aforementioned ability to restrict the scope of Verrou's instrumentation can be used to implement a form of bisection in order to find out which parts of the program are the most unstable. In order to automate such a search, the verrou_dd utility implements in the Verrou ecosystem several variants of the Delta-Debugging algorithm [START_REF] Zeller | Why Programs Fail[END_REF][START_REF] Zeller | Simplifying and isolating failure-inducing input[END_REF]. These algorithms differ in subtle ways in the definition of "unstable" program parts, but it is enough in the following discussion to assume that "unstable" program parts are parts of which the perturbation is correlated to large changes in program results. The full details are given in [START_REF] Févotte | Debugging and optimization of HPC programs in mixed precision with the Verrou tool[END_REF].

Setting up a Verrou environment

Verrou is an open-source tool. The simplest way of installing it for general use is to download and build a combined stable release of Valgrind and Verrou. These are available from the "Release" tab of the GitHub page of Verrou.

As discussed in section 1.3, it is very useful, though not mandatory, to have debugging informations available for the application under study. This can be done by installing debug information packages for all numerical libraries used by the program, and by compiling the program itself with debugging information. In the case of CMake-based packages like ACTS, the latter can by done by using a CMAKE_BUILD_TYPE=Debug or CMAKE_BUILD_TYPE=RelWithDebInfo build configuration.

As a last setup step, it is useful to prepare a standard Verrou exclusion file. As explained in section 1.2, this exclusion list should arrange for Verrou to leave at least the standard mathematical library (libm) unperturbed.

Analyzing the numerical stability of ACTS 2.1 Instrumenting the ACTS test suite

Verrou's random rounding mode was used to evaluate the numerical stability of the ACTS codebase. This was done by running the ACTS unit test suite and integration tests in Verrou, using random rounding mode, and checking which tests would fail and why.

Most of the observed test failures emerged from dubious floating-point data handling practices in the test suite itself. For example, some unit tests would exactly compare the output of a numerical computation to an expected result (as on fig. 1), and could thus only succeed if ACTS' implementation exactly matched the test code. These exact comparisons were largely replaced with approximate comparisons, which decouple the test suite from the implementation and can encode precision expectations more accurately, except in areas such as serialization where exact reproducibility would truly be expected. Similar issues included relative comparison of floating-point data with zero (which only succeeds if the input data is exactly zero) and unrealistic amounts of significant digits in textual data dumps.

Findings in the core ACTS library

Some test suite issues could inform future developments in ACTS' core library. For example, it was found that chaining affine transforms (as done when modeling these transforms as a combination of a translation and Euler rotations) could be very unstable on a numerical level. It could therefore be better to precisely compute the full transform matrix and pass it directly as a program input. This is quite intuitive: in the presence of large translations, a small error on a rotation angle will lead to a large difference in the output of an affine transform.

Additional issues were also found in the core ACTS library:

• one part of the codebase used to compute integer powers of two using the floating-point pow function, where a bit shift would have been faster and more accurate, • a few parts of the codebase would perform divisions whose denominators could get arbitrarily close to zero, without handling the numerical instability that ensues,

• another function would check the azimutal angle difference between two 3D vectors in Cartesian coordinates by converting both vectors to spherical coordinates, subtracting the spherical coordinates, and wrapping the resulting angle in the [-π; π[ range, when that could be done in a more efficient and precise way by leveraging vector product identities.

Edge cases and limits of this approach

One might expect that use of random rounding would lead to non-reproducible test failures. This occasionally proved true, especially in cases where the rounding of a single operation (rather than an accumulation of results) played an important role. However, Verrou comes well-prepared to handle such situations. It provides various alternatives to fully randomized rounding, such as the deterministic rounding modes discussed in 1.1 and an option to control the random number generator seed. These features eliminate reproducibility issues in direct usage of Verrou, and reduce their occurrence in delta-debugging. Overall, the ease with which Verrou allowed numerical stability issues to be discovered, located, and addressed is appreciable. However, the tool did also prove to have some false positives and prerequisites.

Several false positives occured on trigonometry, not just due to the libm issues described above, but also because approximate trigonometry features many edge effects, where a tiny rounding perturbation can make a great difference in the output. This manifested as angles suddenly jumping from -π to π, breaking tests of binned spatial data structures; or as sines and cosines going beyond their valid range of [-1; 1] by a tiny amount, leading NaNs3 to be produced when this data was passed to reciprocal functions.

When it comes to prerequisites, it quickly became clear that although delta-debugging can promptly narrow down some classes of issues, a detailed test suite that can precisely report the cause of failures and the numerical values involved remains an invaluable asset when the instability occurs in a utility function that is called in many different contexts.

Evaluating the viability of reduced precision

Challenges of reduced precision studies

After validating the correctness of ACTS' floating-point computations using Verrou, the next step was to evaluate the usability of single-precision computations in an ACTS context.

As discussed in the introduction, there is a strong efficiency incentive for using reduced floating-point precision whenever it is applicable. Sadly, like many physics libraries, the ACTS codebase was written to use double-precision everywhere as a safe default, with no easy way to configure another floating-point type. Modifying ACTS to make its floatingpoint precision configurable, even without suppressing the assumption that the precision be the same everywhere, is therefore a deep change in the codebase affecting most function signatures and thousands of lines of code. Such a patch is hard to maintain over extended periods of time in such a fast-moving codebase, which led a previous attempt at extending use of single-precision computations in the ACTS codebase to fail.

Fortunately, however, this is an area where dynamic program instrumentation can help. Using the newly introduced reduced precision computation mode of Verrou, it became possible to do a large fraction of the single-precision ACTS validation studies without changing a single line of code in the core ACTS library. Unit tests failures could easily be studied, and "trivial" failures that emerged from unrealistic test expectations (such as expecting single electron-volt resolution on the energy of output particles from a tera-electron-volt collision) could easily be corrected at the test suite level.

Although this part of the study is still ongoing, it already produced some interesting results that will be discussed in the remainder of this paper.

NaN backtraces: A case study

Verrou's new ability to locate occurences of NaN proved particularly useful as it helped discovering a subtle bug that went unnoticed during double-precision validation.

ACTS allows use of interpolated magnetic field maps, which by definition are based on weighted averages of tabulated magnetic field data. Some data on the edge of this magnetic field table used to be left uninitialized, which was not detected because it always had zero weight in the interpolation. However, the property 0 × x = 0 does not always hold when x is an IEEE-754 floating-point number, owing to the fact that 0 × NaN = NaN and 0 × ±∞ = NaN.

Since the probability of random uninitialized data being equal to NaN or ±∞ is much higher for single-precision numbers than for double-precision numbers ( 1128 instead of 1 1024 , owing to the reduced exponent range), this resulted in the appearance of NaN in the output of the single-precision magnetic field interpolation test. The failure could be easily detected and understood using Verrou's new NaN debugging feature (fig. 2). Figure 2. NaN backtrace originating from the use of uninitialized data in ACTS' interpolated magnetic field map. The unabriged stack trace is 42 frames long and goes through complex layers of abstraction: manually locating the point where a NaN is generated would have been a significant undertaking.

ead10b25085b:~/acts-core/spack-build # valgrind --tool=verrou --rounding-mode=float \ > --exclude='pwd'/excludes.ex --demangle=no \ > --trace-children=yes --num-callers=50 \ > ctest -V -R

Looking ahead

Being able to quickly detect, understand, and resolve such issues without maintaining a large ACTS patch means that more time is available to focus on the fundamental issues raised by the use of single-precision FP numbers in HEP's demanding numerical environment. For example, test failures in single-precision mode are currently under investigation in the ACTS particle propagation algorithm, which integrates the equations of motion to follow particle track hypotheses throughout the particle detector in order to evaluate the quality of these hypotheses. This algorithm operates under numerical constraints which lie at the edge of the abilities of single-precision computations (e.g. locating muons with a precision of a fraction of a milimeter after propagating them through the multi-meter radius of the ATLAS detector), and it is possible that at least some parts of it may need to continue operating in double precision or use a compensated algorithm.

Verrou's delta-debugging abilities will therefore be used to understand if ACTS' particle propagation code features specific "precision bottlenecks" which could be made to work in single precision through algorithmic improvements or focused use of extended precision. Should that turn out not to be the case, the ACTS track propagation codebase will need to keep using double precision internally.

Conclusion Summary

Future HEP experiments will face unprecedented data processing challenges. Opening up established particle tracking codebases to new users, extending their validation procedure to maximize their ability to face new constraints, and optimizing them to help them better utilize available computing resources, are all worthwhile activities in this perspective.

In this context, the Valgrind-based Verrou tool was used to stress-test the ACTS codebase with new forms of numerical validation, and is now being used to extend usage of reducedprecision arithmetic throughout this codebase. One goal of this study is that in the future, ACTS developers will be able to confidently use single-precision arithmetic as a safe default, validate the stability of the results through random rounding, and only fall back to less efficient double-precision arithmetic where it is truly necessary.

Future work

Most of the ACTS improvements that were discussed in part 2 of this study have been merged into the ACTS codebase. As discussed in part 3, the focus of this project is now on planning out a reduced-precision port of ACTS which will rationalize choices of floating-point precision throughout the codebase.

Beyond ACTS, this study revealed some future areas of improvement in the Verrou toolchain, and in particular its delta-debugging features. The functionality is currently less generally applicable than it could be, owing to the fact that it does not handle utility libraries well. It is planned to improve upon this situation in 2019 by adding call site path sensitivity to Verrou, allowing the tool to report in which caller context instrumenting a function is detrimental, and to do delta-debugging on stack traces rather than mere symbols.

Parallelization of the delta-debugging process would be another desirable feature, owing to the fact that use of Verrou serializes and slows down program execution significantly, and that delta-debugging needs to test many independent program configurations. This feature is already available in an experimental form, but its finalization is still some way off.

Figure 1 .

 1 Figure 1. ACTS unit test failure originating from exact comparison of floating-point results

As produced for example by using the -g command-line switch with gcc.

Not-a-number, an error code produced by IEEE-754 arithmetic when performing meaningless computations

This project has received funding from the European Union's Horizon 2020 Research and Innovation programme under Grant Agreement no. 654168.


François-Xavier Coudert
email: fx.coudert@chimie-paristech.fr

Anne Boutin

Alain H Fuchs
email: alain.fuchs@cnrs-dir.fr

A Thermodynamic Description of the Adsorption Induced Structural Transitions in Flexible MIL-53 Metal-Organic Framework

We briefly review the concept of Temperature-Loading phase diagram that we derived from an osmotic statistical ensemble analysis of the adsorption induced structural transitions observed in the MIL-53 family of metal-organic framework nanoporous materials. We highlight the generic nature of this diagram and comment on the occurrence of breathing transitions depending on the guest fluid, the framework functionalization and the nature of the metal centre.

Introduction

Among the wide class of Metal-Organic Frameworks (MOFs) and other related open framework hybrid materials, the subclass of Soft Porous Crystals (SPCs) has attracted a lot of attention in the past few years. According to Kitagawa, who introduced this notion, "soft porous crystal are defined as porous solids that possess both a highly ordered network and structural transformability. They are bistable or multistable crystalline materials with long range structural ordering, a reversible transformability between states and permanent porosity". 1 It has been recently shown that the "softness" of these materials, which was somewhat ill-defined up to now, actually stems from a highly anisotropic mechanical (or elastic) behaviour, with some directions of the crystal exhibiting very low Young's modulus and shear modulus. 2 SPCs exhibit large-amplitude structural deformations induced by various external stimuli such as temperature, mechanical pressure, or guest adsorption. Guest-induced structural transitions include the gate adsorption and the breathing phenomena. Gateadsorption typically involves an abrupt structural transition between a nonporous state and a porous crystalline host that is induced by gas adsorption. 3,4 The breathing phenomenon consists of two successive adsorption-induced crystal-to-crystal transformations, from a large pore (lp) state to a narrow pore (np) state, and back again to the lp structure. 5,6 In a series of articles, the present authors have investigated the equilibrium thermodynamics of adsorption-induced and pressure induced breathing transitions using an analytical approach based on the osmotic statistical ensemble. Some key questions were answered, such as the prediction of the occurrence or absence of guest-induced breathing for a given guest-host system, 7,8,9 and the successful derivation of a {temperature-gas pressure} phase diagram for the several gases (xenon, methane, carbon dioxide)/MIL-53 systems 10,11,12 as well as breathing phase diagram for binary mixtures of methane and carbon dioxide/MIL-53. 13 The physical mechanism of the breathing transitions was addressed by viewing the adsorption-induced stress exerted on the material as the stimulus that triggers breathing transitions, in a way similar to lp-tonp structural transitions can be induced by mechanical compression. This new view of the breathing that we proposed implies that the structural transitions in SPCs takes place when the stress reaches a certain critical threshold. 14,15 Though the exact chemical origins of this stress have not yet been fully uncovered, it is similar in nature to the elastic limit (or yield strength) in classical mechanics, where the deformation stops being quasistatic. For example, a recent molecular simulation study has shed some light on the pressure-induced amorphization of Zeolitic Imidazolate Framework ZIF-8: 16 the amorphization transition in this material was shown to be the result of pressure-induced softening, until the pressure reached a value at which the crystal was not mechanically stable (Born's stability criteria no longer hold). Although it has not been equivocally demonstrated, we envision that a similar mechanism might be in play in the breathing of MIL-53. This hypothesis of a "critical stress threshold" enabled to explain, at least qualitatively, the hysteresis phenomenon observed in the breathing transitions: when approaching the structural transition, the material is in two different phases on the adsorption and desorption branches, whose critical stress need not be equal (it was demonstrated, for example, that other mechanical properties such as the bulk modulus vary between the two phases 15 ). Moreover, this concept was used as a basis for building a multiscale dynamic model in order to capture the dynamics of adsorption-induced deformation and structural transformation in MIL-53 based on the coupling of the host-guest adsorption interactions with the elastic response of the three-dimensional framework of a given geometry. 17,18 Finally the issue of multistability and intermediate structures was recently addressed, using the osmotic potential approach. 19 In this paper we briefly review the equilibrium Temperature-Loading phase diagrams (i.e. the {temperature-gas pressure} diagrams) that were computed for different guest-MIL-53 systems. We highlight the generic nature of this diagram and comment on the occurrence of breathing transitions depending on the guest fluid, the framework functionalization and the nature of the metal centre.

Temperature-Loading diagrams for several guest-MIL-53(Al) systems

We have used a model based on the so-called osmotic ensemble, 7 which is the appropriate statistical ensemble to describe fluid adsorption in a flexible porous material. Since the MIL-53 material exhibits clear structural transitions between two metastable framework structures, namely the "narrow pore", np phase and the "large pore", lp phase (see Figure 1), we were able to use a simplified osmotic sub-ensemble to describe the equilibrium between host structures upon fluid adsorption. 7,8 Langmuir fits of the experimental isotherms served as approximations to the rigid host isotherms in both the lp and np structures. We showed that, at a fixed temperature, the occurrence of breathing (or other structure transitions) upon adsorption could be rationalized by three key parameters with clear physical meaning: the affinity of the fluid for each phase (as described by the Langmuir's Henry constant); the capacity (or saturation uptake) of fluid in each phase; the free energy difference between the empty phases. In the first study of this series, performed on the {Xenon-MIL-53 (Al)} system, 10 we have used stepped isotherms at two temperatures to determine the transition enthalpy and entropy of the empty host material, as well as the free energy difference between the empty lp and the np structures. Not unexpectedly, the lp form was predicted to be the most stable one at room temperature, while the np structure becomes the most stable one below 203 K. One of the interests of the osmotic thermodynamic model is that it enables to compute equilibrium thermodynamic data for the bare host material using thermodynamic adsorption data only. The lp phase was found to be stable at high temperature and again at lower temperature.

There is an intermediate np phase stability domain for xenon pressure lower than a limiting pressure of around 1.6 bar. This result is reminiscent of the re-entrant behavior observed in some liquid crystals. [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] As noted above, however, the low-temperature stable phase at zero pressure (empty material) is the np phase.

The phase diagrams for the {CO 2 , MIL-53(Al)} and {CH 4 , MIL-53(Al)} are also shown in Fig. 2. 12 All three diagrams have in common the equilibrium np-lp temperature T 0 of 203 K at zero pressure. The initial slope of the transition curve is proportional to the logarithm of (K np /K lp ), the ratio of adsorption affinities in the two structures, and is thus strictly positive since the affinity of the guest adsorbate for the closed form of the framework is higher than for the open form. 7,10 The condition K np /K lp >1 thus favors the closed np phase, and consequently the phase transition temperature increases with the gas loading (i.e. the stability domain of the np phase increases with P adsorbate ). At higher temperature, the transition free energy increases, and it becomes more and more difficult to maintain the np form as the most stable one. This causes the observed bending of the transition line. For obvious entropy reasons, the lp phase will eventually become more stable at high temperature, regardless of the gas loading. This situation is also true at high pressure. As the adsorbate pressure increases, at any temperature, the lp structure will eventually become more stable than the np one because it can accommodate a higher loading of guest molecules. Since the lp phase is the most stable one at high enough temperature as well as at high adsorbate pressure, one has to conclude that the stability domain of the np phase should be limited in adsorbate pressure (P max ~ 5 bar in the case of carbon dioxide, 2 bar in the case of methane and 1.6 bar in the case of xenon, see Figure 2).

The above thermodynamic considerations are very general and obviously not limited to the special case of one or the other gas adsorption. The condition K np / K lp > 1 is expected to hold true for all the simple guest molecules that have been investigated so far. This means that there should be a range of temperatures above the equilibrium nplp transition temperature of the bare MIL-53(Al) material (203 K in our model, subject to an estimated uncertainty of ±10 K) where the initially empty open structure contracts upon guest molecule adsorption. The fact that this has not been observed in some cases at room temperature simply means that the transition line maximum in this system is below the room temperature. The present findings for methane clearly confirm these predictions.

The difference in the stability domain of the np phase in the three different cases can be qualitatively understood as follows. For each system, there is a temperature T max (see Figure 3a), above which gas adsorption does not induce the lp->np phase transition anymore. Since the driving force for the closure of the lp structure is the guest-host interaction, which induces the cell contraction, one may simply write: k T max ~ ∆H ads,lp .

The increase in T max in going from CH 4 to Xe and CO 2 can then simply be explained by the increase in adsorption enthalpy in the lp phase. While P max is approximately the same for CH 4 and Xe, its value for CO 2 is larger by a factor of ~3. This difference can be accounted for by a larger interaction energy between guest molecules in the case of carbon dioxide, due their quadrupole moments. This causes a stronger ordering of the CO 2 molecules in the np phase and increases the stability domain of this phase.

In figure 3b we sketch the three phase diagrams in reduced units of (T/T max ) and (P/P max ). Some sort of a "universal" phase diagram seem to come out of this presentation. This confirms the existence of a generic temperature-loading phase diagram, as long as the adsorbate is a simple gas such as methane, carbon dioxide or xenon. It must be recalled at this stage that the osmotic model predicts the conditions of thermodynamic stability at full equilibrium and does not take into account hysteresis effects. Hysteresis was systematically encountered in all reported MIL-53 experiments and often leads to some complicated mixtures of phases. [START_REF] Salles | [END_REF] This model describes what would happen in a homogeneous system at full equilibrium. The osmotic model is aimed at describing the thermodynamics behind the scene. The hysteresis issue is being addressed in some of our other studies. [14][15][16][17][18][19] Finally one should mention that the osmotic model was extended to predict the behaviour of fluid mixtures adsorbing in flexible nanoporous materials by proposing an analytical scheme called OFAST (Osmotic Framework Adsorbed Solution Theory). 13 Figure 4 represents the {CO 2 /CH 4 , MIL-53(Al)} diagrams for different values of the carbon dioxide mole fraction. We can see that, in addition to the re-entrant nature of the phase diagrams themselves in (T, P) space, there is also a nonmonotonic evolution of the maximal temperature and pressure of the np phase as a function of the mixture's composition. This unexpected result, first hypothesized on a purely theoretical thermodynamic basis, 9 was later proven true by experimental measurements. 13 An even more drastic change in the MIL-53 composition consists in replacing the aluminium center by a gallium atom. MIL-53 (Ga) was synthesized and characterized by the group of Férey, 22 and then further investigated by Boutin and coworkers. 23 Breathing transitions were found to take place upon several gas adsorption 22 but the stable phase at room temperature was found to be the np narrow pore structure. 22,23 A recent quantum chemistry investigation by Coudert and coworkers allowed to understand that np phase was more stable in the case of MIL-53 (Ga), because of the more diffuse orbitals of Ga compared to Al. 24 Thus the np phase is the stable structure at room temperature (rather than lp as Al-MIL53). The effect of replacing Al by Ga is simply to increase the value of T 0 , the np-lp equilibrium phase transition temperature of the empty MIL-53 framework. The whole Temperature-Loading phase diagram is thus

Conclusion

The temperature-loading phase diagram of the flexible MIL-53 family of materials is a useful tool to understand the behaviour of adsorption induced breathing structural transitions. Whether or not these transitions take place upon adsorption of a particular gas or gas mixture depend on the relative location of the stability regions of the np and lp states in the system's (T, P, x) space, where T is the temperature, P is the adsorbate gas pressure and x is a mole fraction describing the gas composition. Even though this is just an equilibrium thermodynamic view of the breathing process, which means for instance that it lacks the hysteresis effects that take place in the experiments, this temperature-loading concept has allowed to demonstrate that the breathing effect in MIL-53 is a very general phenomenon, that should be observed in some region of the (T, P) space whatever the guest molecule, since it is expected that the affinity of an adsorbate for the closed np form of the framework will always be higher than for the open lp structure. We also believe that this very general model can provide a useful tool for experimentalists to tune their experimental conditions. As such, the phase diagrams described here and the thermodynamic models developed to build them naturally complement the other approaches used in the field of flexible porous materials, namely poromechanics 25,26 (at the macroscopic scale) and atomistic molecular simulation (at the microscopic scale); [START_REF] Ghoufi | [END_REF]28,29,30 the advantages and downsides of all three methods were recently reviewed in ref. 31.

Figure 1 .

 1 Figure 1. Representation of the metastable lp (top) and np (bottom) structures of the MIL-53(Al) material, as a 2 × 2 × 2 supercell viewed along the axis of the unidimensional channels.

  The osmotic model, with the use of the fits performed on the experimental adsorption isotherms, enables to investigate the full temperature-loading phase diagram of {Xe, MIL-53(Al)}. By solving the osmotic thermodynamic equations numerically, we determined for each temperature, whether breathing takes place and, if so, what the transition pressures are. The predicted Temperature-Xenon gas Pressure diagram is shown in Figure2.

Figure 2 .

 2 Figure 2. Comparison of the Temperature-gas Pressure phase diagrams for CO 2 (red), CH 4 (green) and Xe (blue) adsorption in MIL-53 (Al). 12 Experimental transition pressures are indicated by square symbols.

Figure 3 .

 3 Figure 3. Left: Comparison of the temperature-vapour pressure phase diagrams for CO 2 (red), CH 4 (green), and Xe (blue) adsorption in MIL-53(Al), with the extremal temperatures and pressures marked. Right: same three phase diagrams in reduced units of (T / T max ) and (P / P max ). Left figure is reproduced with permission from ref. 12, © 2010 American Chemical Society.

Figure 4 . 3 .

 43 Figure 4. Temperature-pressure phase diagram of MIL-53(Al) upon adsorption of a CO 2 /CH 4 mixture, with increasing CO 2 molar fraction. Dashed lines correspond to pure component diagrams: CO 2 in green, CH 4 in blue. Reproduced with permission from ref. 13, © 2011 American Chemical Society.

Figure 5 .

 5 Figure 5. Temperature-CO 2 pressure phase diagrams for regular MIL-53(Al) (red) and amino-MIL-53(Al) (blue) materials. The T 0 are the equilibrium phase transition temperature between the lp and np phases at zero pressure (i.e. the empty material) for both materials. Modified with permission from ref. 11, © 2010 Elsevier Inc.

Acknowledgements

This short review paper is dedicated to Professor Pierre Turq, on the occasion of his 70th birthday. Pierre was a pioneer in introducing statistical mechanics and molecular simulations at all scales in France, in the field of chemistry and physical chemistry, and the French community owes him a lot.

The authors acknowledge funding from the Agence Nationale de la Recherche under the project ''SOFT-CRYSTAB'' (ANR-2010-BLAN-0822).


François-Xavier Coudert
email: fx.coudert@chimie-paristech.fr

Alain H Fuchs

Alexander V Neimark
email: aneimark@rutgers.edu

Adsorption Deformation of Microporous Composites

We study here the behavior of flexible adsorbent materials, or Soft Porous Crystals, when used in practical applications as nanostructured composites such as core-shell particles or mixed matrix membranes. Based on simple models and the well-established laws of elasticity, we demonstrate how the presence of a binder results in an attenuation of the adsorption-induced stress and deformation. In the case where the adsorbent undergoes adsorption-induced structural transitions, such as the gate opening phenomenon occurring in some metal-organic frameworks, we show that the presence of the binder will result in shifts of the adsorption-induced transition pressures.

 Most prominent are gate

opening and breathing transitions, which have been explored in great detail by using in situ XRD scattering drawing on a variety of metal-organic framework (MOF) materials. 19,20,10 Microscopic mechanisms of MOF flexibility have been examined by detailed first principles and atomistic molecular dynamic and Monte Carlo simulations performed on the crystal cell level. 21,22,23,24,25 A rigorous thermodynamic approach 11 has been proposed to predict the adsorption stress from adsorption isotherms. 26 However, these studies are limited to single adsorbent particles, while the vast majority of practical applications require the use of composite materials, 27 such as micrometer size adsorbent particles embedded into a permeable matrix 28 , 29 , 30 or thin films anchored on a support. 31,32 Although composite MOF materials have been studied from the point of view of adsorption and gas separation, such as e.g. MOFs coated with mesoporous silica, 33 there has been so far no specific study focused on flexibility, which is still a completely open question.

In this letter, we explore the flexibility of composite materials formed by a soft porous crystal and a non-adsorbing matrix. In particular, we study the extent to which the elastic properties of the non-adsorbing matrix affect the adsorption-induced deformation of the adsorbent particles and the composite as a whole. Using a simple yet instructive spherical core-shell model, we find that adsorption deformation in composite is reduced compared to unconfined particles and the magnitude of this effect is determined by the both elastic moduli of the matrix and the morphology of the nanostructured composite. The reduction of adsorption deformation leads to the shift of the positions of morphological transformations, which can be even suppressed due to matrix rigidity. These conclusions have important implications on the feasibility assessment of the potential applications, which are based on the effects of adsorbent flexibility. For example, widening the hysteresis loops in adsorption-desorption behavior can improve the amount of mechanical energy stored in a high-pressure liquid intrusion. 23 Tuning the flexibility of the As a model system, let us consider a composite spherical particle made of an adsorbing core of radius R 1 (in our case, a particle of soft porous crystal) and non-adsorbing elastic shell of external radius R 2 . Both adsorbent and matrix materials are assumed isotropic. The elastic properties of the composite are characterized by the volumetric modulus K a of the adsorbent, whose deformation is considered uniform, and by the Young modulus E and Poisson's ratio ν of the matrix. Under the action of adsorption-induced stress 𝜎𝜎 " , the adsorbent core is deformed uniformly with linearly progressing radial displacement 𝑢𝑢 $ and constant radial strain 𝑢𝑢 $$ given by

𝑢𝑢 $ = & ' () * +, ' 𝑟𝑟 ; 𝑢𝑢 $$ = & ' () * +, ' at r < R 1 ( 1
)
where r is radial coordinate. 35 The external pressure 𝑃𝑃 / acting on the core-shell boundary r = R 1 is unknown and is determined from the solution of the classical Lamé problem of the deformation of the outer spherical shell under action of internal P 1 and external P 2 pressures. 35 The general solution of the Lamé problem implies that in the shell, the radial displacement 𝑢𝑢 $ , strain 𝑢𝑢 $$ , and stress 𝜎𝜎 $$ are given by the following equations:

𝑢𝑢 $ = 𝑎𝑎𝑟𝑟 + 𝑏𝑏/𝑟𝑟 4 , 𝑢𝑢 $$ = 𝑎𝑎 -2𝑏𝑏/𝑟𝑟 + , 𝜎𝜎 $$ = 8 /(49 𝑎𝑎 - :; *<= > $ ? , at R 1 < r < R 2 . ( 2
)
The constants a and b are determined from the boundary conditions,

𝜎𝜎 $$ 𝑅𝑅 / = -𝑃𝑃 / (3)
𝜎𝜎 $$ 𝑅𝑅 4 = -𝑃𝑃 4 , (4) as 𝑎𝑎
Note that the composite morphology is characterized by the ratio g of the adsorbent and matrix volumes,

𝑔𝑔 = A * ? A : ? (A * ? (8)
The unknown pressure 𝑃𝑃 / is determined from the third boundary condition of equality of the displacements in the core and in the shell at r = R 1 , namely:

𝑢𝑢 $ (𝑅𝑅 / ) = (𝜎𝜎 " -𝑃𝑃 / )𝑅𝑅 / /3𝐾𝐾 " = 𝑎𝑎𝑅𝑅 / + 𝑏𝑏/𝑅𝑅 / 4 (9)
Combining condition ( 9) with ( 5) and ( 6) one arrives at a linear equation for the boundary pressure 𝑃𝑃 / , which solution gives:

𝑃𝑃 / = & ' H ? *<M N O( *PM Q , ' ) : /H, ' M Q H ? *<M N O (10)
As such, the sought radial strain in the adsorbent core is given as

𝑢𝑢 $$ = (𝜎𝜎 " -𝑃𝑃 / )/3𝐾𝐾 " = & ' +, ' - & ' /+, ' H(/(R)(//IO(//+,)) : /H, ' (R/,H+(/(R)/IO) (11)
Equation 11 shows that the adsorption-induced strain 𝜎𝜎 " is reduced by the elastic confinement, since the second term on the right-hand side is guaranteed to be negative. The extent of the lowering or "damping" of the deformation upon adsorption depends on both the geometrical characteristics of the composite material (through the ratio g) and the mechanical properties of the elastic shell (moduli K and µ). It should be noted that the volumetric strain 𝜖𝜖, which is the quantity involved in thermodynamic descriptions and models of adsorption deformation, is equal to three times the radial strain 𝑢𝑢 $$ , 𝜖𝜖 = 3𝑢𝑢 $$ . As such, Eqn. 11 can be rewritten in terms of the volumetric strain 𝜖𝜖 as a function of the adsorption stress 𝜎𝜎 " and external pressure, 𝑃𝑃 4 = 𝑃𝑃 ext :

𝜖𝜖 = & ' , ' - & ' /, ' H(/(R)(//IO(//+,)) ext /H, ' (R/+,H(/(R)/IO) (12)
We will now consider specific cases, schematized in Figure 1, that are relevant for practical applications of composite materials containing soft porous crystals.

Let us first consider an unconfined adsorbent under external pressure 𝑷𝑷 e ex xt t . This case corresponds to 𝑃𝑃 4 = 𝑃𝑃 XYZ and 𝑔𝑔 → ∞ that reduces to the standard equation

𝜖𝜖 = (𝜎𝜎 " -𝑃𝑃 ext )/𝐾𝐾 " (13)
This is the equation used in thermo-mechanical models of adsorption-induced deformation of soft porous crystals when the pure MOF is considered. 26 In the absence of external mechanical pressure (P ext = 0), it further reduces to the simple Hooke's law 𝜖𝜖 = 𝜎𝜎 " /𝐾𝐾 " , which will in the following serve as our baseline reflecting the behavior of flexible MOFs in the absence of confining phase.

In the case of an adsorbent particle in infinite elastic matrix, the volume of the exterior matrix is much larger than that of the crystal. It is a very simplified version of real composite materials used in practical applications, e.g. such as a mixed matrix membrane with particles of soft porous crystal embedded in a surrounding polymer phase.

This case corresponds to 𝑃𝑃 ext = 0 and 𝑔𝑔 → 0. The volumetric strain from Eqn. 12 is now equal to

𝜖𝜖 = & ' , ' 1 - / /H, ' /IJ = & ' , ' × / /HIJ/, ' < & ' , ' (14)
From this equation, we see that the deformation of the material is reduced, by a constant factor that depends on the relative elasticity of the adsorbent and matrix: the stiffer it is, the more it prevents the core particle from changing its volume. The adsorption-induced stress is dampened, or diminished, by a factor of / /HIJ/, ' .

Another case of interest is that of core-shell nanoparticles, i.e. the general case in which g depends on the shell size but 𝑃𝑃 ext = 0. This geometry of a nanoparticle of MOF crystal coated (or surrounded) by a non-adsorbing matrix is particularly relevant for applications, where the shell could be used to reinforce the chemical and mechanical stability of the MOF core. For that case, the adsorption-induced strain is:

𝜖𝜖 = & ' , ' 1 - / /H, ' M ?Q H(/(R)/IJ < & ' , ' (15)
Like in the previous case, the strain is reduced compared to the unconfined particle, but this time the factor depends on both the bulk (volumetric) K and shear µ moduli of the shell matrix, as well as the geometry (through g). For the specific case of a thin shell of thickness h (a coating of the MOF), this factor equals 𝑔𝑔 = 𝑅𝑅 / /3ℎ.

Finally, now turn to a composite material (soft porous crystal core and elastic shell) jacketed by a non-deformable outer container. In practice, this represents a composite that is surrounded by a much stiffer material. This case thus corresponds to different boundary conditions, because the outer stress P 2 is undetermined, but the volume is conserved leading to a condition of zero displacement at 𝑟𝑟 = 𝑅𝑅 4 , 𝑢𝑢 $ 𝑅𝑅 4 = 0 . Solution of the Lamé equations with this boundary condition gives the following equation for the volumetric strain (see mathematical derivation in Appendix):

𝜖𝜖 = & ' () * , ' = & ' , ' × / /H MQ Q ' H Na *<M ?Q ' < & ' , ' (16)
Hence, despite the different nature of the boundary conditions, the end result is similar: additional constraints on the soft porous crystals cause further reduction in volumetric strain.

Finally, let us highlight the effect of the confinement-induced dampening of the adsorption deformation drawing on the example of a stimuli-responsive MOF. We consider here a MOF crystal undergoing a gate opening phenomenon, i.e. the adsorption-driven opening from a nonporous closed structure (or structure of lower pore volume) into a microporous open phase (or structure of large pore volume). Following the thermo-mechanical view (or "stress model") of adsorption-induced structural transitions in soft porous crystals, 26 gate opening occurs upon adsorption when the outwards stress exerted by the guest molecules reaches a certain critical threshold, the limit of stability of this closed phase. Symmetrically, upon desorption the transition occurs when the adsorption-induced stress falls below another threshold, which is the limit of To illustrate the impact of the external confinement on the gate-opening phenomenon, we consider a crystal in infinite elastic matrix. The adsorption stress isotherm (stress vs. gas pressure) and the respective adsorption isotherm are plotted by dashed lines in Fig. 2. One can see that the changes in the adsorption stress lead to a shift in the position of the structural transition. In the example given in Fig. 2, this phenomenon leads to a widening of the hysteresis loop, with gate opening occurring at a higher pressure and gate closing at a lower pressure. It could also be possible, depending on the nature and shape of the adsorption stress profiles, to observe a reverse effect -e.g., with gate closing at a higher pressure than in the unconfined soft porous crystal, or gate opening at lower pressure. Similar conclusions can be made for the case of core-shell particles.

In conclusion, we have shown that the elastic confinement present in composite materials affects adsorption-induced deformation of soft porous crystals. This effect is of particular interest, since flexible adsorbents in practical applications are used as composites such as core-shell particles or mixed matrix membranes. By deriving the mechanical equations for the deformation of model composite materials, we show that this effect depends on the geometric nature of the composite as well as on the respective elastic properties of the adsorbent (through its bulk modulus) and the elastic matrix (through its bulk and shear moduli). In all cases, however, the presence of the binder results in an attenuation of the adsorption-induced deformation. In the case where the adsorbent undergoes adsorption-induced structural transitions, such as the gate opening phenomenon occurring in metal-organic frameworks, the presence of a binder around the adsorbent can result in shifts of the transition pressures. In future works, it will be of interest to expand the study from the radial stress to the tangential stress at the crystal-binder interface. This tangential stress cannot exceed the limit of material integrity of the elastic medium that 10 determines the conditions, at which the nanostructured composite "breaks down", i.e. contact between the soft porous crystal and the surrounding elastic medium is lost. Noteworthy the analysis performed in this work is limited to isotropic crystals; its extension to anisotropic structures is highly desired yet involves a much more elaborated treatment of adsorption deformation. We hope that this first theoretical treatment of the open question of flexibility of MOF-based nanocomposite materials will spur experimental research in that direction.

Figure 1 .

 1 Figure 1. (a) Model system representing a composite material. (b) Three special cases of the

  The volumetric 𝐾𝐾 and shear 𝜇𝜇 moduli are used in the right-hand side of Eqns. 5 and 6 instead of Young's modulus E and Poisson's ratio ν, to which they are related in an isotropic elastic medium by:

  stability of the open phase. This mechanism results in step-wise adsorption-desorption isotherms forming a hysteresis loop exemplified in Figure2(solid lines), taking for the adsorption stress profile a Langmuirian equation, as derived in Ref.26.

Figure 2 .

 2 Figure 2. Gate-opening phenomenon in an isolated crystal (solid lines) and in a crystal

Acknowledgments. FXC acknowledges PSL Research University's financial support through projects DEFORM and MECADS (grant ANR-10-IDEX-0001-02). AVN acknowledges support from the Rutgers NSF ERC "Structured Organic Composite Systems".

Appendix A. Derivation of adsorption-induced deformation in a jacketed composite

Let us consider a composite material jacketed by a non-deformable outer container. The boundary condition in this case is

We start from the general solution to the Lamé problem in the shell region as given by Equation ( 2), which we re-write as

The boundary conditions are and for the strain in the adsorbent core 𝑢𝑢 $$ = (𝜎𝜎 " -𝑃𝑃 / )/3𝐾𝐾 " = 𝜎𝜎 " /3𝐾𝐾 " (1 + 𝑔𝑔𝐾𝐾/𝐾𝐾 " + 4𝜇𝜇(1 + 𝑔𝑔)/3𝐾𝐾 " ) (A.8) 𝜖𝜖 = (𝜎𝜎 " -𝑃𝑃 / )/𝐾𝐾 " = 𝜎𝜎 " /𝐾𝐾 " (1 + 𝑔𝑔𝐾𝐾/𝐾𝐾 " + 4𝜇𝜇(1 + 𝑔𝑔)/3𝐾𝐾 " ) (A.9)


François-Xavier Coudert
email: fx.coudert@chimie-paristech.fr

Anne Boutin

Alain H Fuchs

Alexander V Neimark
email: aneimark@rutgers.edu

Adsorption Deformation and Structural Transitions in Metal-Organic Frameworks: From the Unit Cell to the Crystal

Keywords: adsorption, adsorption stress, structural transitions, porous materials, breathing, metal-organic frameworks. adsorption or temperature or stress

Much attention has recently been focused on Soft Porous Crystals, a fascinating subclass of Metal-Organic Frameworks that behave in a remarkable stimuli-responsive fashion, presenting structural changes of large amplitude in response to guest adsorption, mechanical pressure or variations in temperature. In this perspective, we summarize the recently developed thermodynamic and mechanical theoretical models developed for the understanding of these materials, based on the simple concepts of adsorption stress and the osmotic thermodynamic ensemble. We show how these models aim at providing a coherent picture of adsorption-induced deformation and structural transitions in flexible Metal-Organic Frameworks, all the way from the length scale of the unit cell to that of the full crystal. In particular, we highlight the new perspectives opened by these models, as well as some of the important open questions in the field.

Much attention has recently been focused on metal-organic frameworks (MOFs), a wide class of designer microporous materials that garner a lot of interest for their potential applications in the fields of separation, catalysis, strategic gas capture and storage, and drug delivery. [START_REF] Zhou | Introduction to Metal-Organic Frameworks[END_REF]2 Soft Porous Crystals (SPCs) represent a specific subclass of MOFs with remarkable flexibility. These materials feature dynamic crystalline frameworks with reversible structural transformations of large amplitude in response to variation of external physical stimuli such as gas adsorption, temperature, or mechanical pressure. [START_REF] Horike | Soft Porous Crystals[END_REF] SPCs display a large gamut of flexibility mechanisms depicted in Figure 1, [START_REF] Coudert | Thermodynamic Methods and Models to Study Flexible Metal-Organic Frameworks[END_REF] including: local framework dynamics, as sorption-induced linker reorientation in ZIF-8; 5 negative thermal expansion, as in the IRMOF family of materials; [START_REF] Dubbledam | Exceptional Negative Thermal Expansion in Isoreticular Metal-Organic Frameworks[END_REF] gradual swelling upon uptake of solvent molecules, as demonstrated in the MIL-88 family; [START_REF] Serre | Role of Solvent-Host Interactions That Lead to Very Large Swelling of Hybrid Frameworks[END_REF] gate opening, which involves an abrupt structural transition between close and open pore structures that is induced by gas adsorption (e.g. in the ELM family); [START_REF] Cheng | Tuning of Gate Opening of an Elastic Layered Structure MOF in CO2 Sorption with a Trace of Alcohol Molecules[END_REF] and the breathing phenomenon, which consists of two successive adsorption-induced crystal-to-crystal transformations, for example in the case of the MIL-53 family of materials, from a large pore (lp) state to a narrow pore (np) state, and back again to the lp state. [START_REF] Serre | An Explanation for the Very Large Breathing Effect of a Metal-Organic Framework during CO2 Adsorption[END_REF] A growing number of SPC structures were synthesized and reported in the literature, and although none of them have yet been directly used at the industrial scale, [START_REF] Czaja | Industrial applications of metal-organic frameworks[END_REF] they have been proposed for a large range of potential practical applications. In addition to the general application of MOFs, flexible frameworks are expected to present an intrinsic interest due to their large-scale stimuli-responsive transformations, [START_REF] Bureekaew | Chemistry and application of flexible porous coordination polymers[END_REF] which could leverage novel nano-biotechnologies, such as sensing for detecting traces of organic molecules, [START_REF] Férey | Hybrid porous solids: past, present, future[END_REF] slow release of drugs for long-release single-injection therapies, [START_REF] Horcajada | Flexible Porous Metal-Organic Frameworks for a Controlled Drug Delivery[END_REF] and specific gas separations. [START_REF] Thallapally | Flexible (Breathing) Interpenetrated Metal-Organic Frameworks for CO2 Separation Applications[END_REF][START_REF] Remy | Modeling the Effect of Structural Changes during Dynamic Separation Processes on MOFs[END_REF] The adsorption-induced deformation and adsorption-induced transitions in flexible MOFs were first addressed in the literature with respect to their structures and energetics, both experimentally and using molecular simulation methods. From the theoretical point of view, there are two key fundamental concepts for understanding the behaviors reported in the literature. First, the adsorption-induced deformation can be viewed as a response of the porous material to an adsorption-induced stress, exerted by the adsorbed fluid on the host solid. [START_REF] Ravikovitch | Density Functional Theory Model of Adsorption Deformation[END_REF] Secondly, a rigorous thermodynamic description of the interplay between adsorption and deformation including structural transitions is provided by the use of the osmotic ensemble, [START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF] a semi-open statistico-mechanical ensemble in which the quantity of solid, temperature and mechanical stress are fixed, while the quantity of guest molecules and the unit cell shape and volume may change. The osmotic ensemble has been used to build theoretical models of the influence of adsorption on the equilibrium between metastable host structures. [START_REF] Coudert | Thermodynamic Methods and Models to Study Flexible Metal-Organic Frameworks[END_REF] Such analytical models are useful in rationalizing the various types of structural transitions observed and the key parameters controlling their occurrence, [START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF] in processing the experimental data, [START_REF] Boutin | Breathing Transitions in MIL-53(Al) Metal-Organic Framework Upon Xenon Adsorption[END_REF] or in making predictions of mixture co-adsorption from pure component data. [START_REF] Coudert | The osmotic framework adsorbed solution theory: predicting mixture coadsorption in flexible nanoporous materials[END_REF] The osmotic ensemble has also been used as a basis for designing molecular simulation methods focused on adsorption in flexible materials. [START_REF] Ghoufi | Hybrid Monte Carlo Simulations Combined with a Phase Mixture Model to Predict the Structural Transitions of a Porous Metal-Organic Framework Material upon Adsorption of Guest Molecules[END_REF][START_REF] Zang | Osmotic ensemble methods for predicting adsorptioninduced structural transitions in nanoporous materials using molecular simulations[END_REF][START_REF] Bousquet | Free energy landscapes for the thermodynamic understanding of adsorption-induced deformations and structural transitions in porous materials[END_REF][START_REF] Santander | Simulating Adsorptive Expansion of Zeolites: Application to Biomass-Derived Solutions in Contact with Silicalite[END_REF] Atomistic molecular simulations enable one to link microscopic details of the host structure and host-guest interactions to macroscopic behavior with an unmatched degree of accuracy. However, they have limitations when it comes to understanding the mechanisms of deformation and structural transitions. One limitation is the length and time scales currently achievable, directly due to the computational cost of the atomistic description: most atomistic simulations range in size between one and a few dozen unit cells, which is far from the actual size of a micrometric (or even submicrometric) crystal. The second limitation is that molecular simulation of porous materials typically use periodic boundary conditions, in order to lower the computational cost and to minimize the finite size and boundary effects, but the periodic boundary conditions do not reproduce the conditions of the sample boundaries and impose artificial constraints on the deformation pathways in the material. This approximation is thus no longer valid for understanding the mechanisms of deformation and structural transitions, and more adequate simulation techniques have to be developed to avoid this issue. Thus, while adsorption-induced deformation and structural transitions are by now well understood from the point of view of thermodynamics and energetics, a mechanical approach to the dynamics of these phenomena has not been developed. In particular, there have been so far no experimental studies of structural transition dynamics at the scale of the monocrystal. These aspects are, however, of large importance. Firstly, experimental data show that structural transitions in SPC are almost always associated with wide hysteresis loops, whether they are triggered by adsorption, temperature, or mechanical pressure. Secondly, recent experimental results from the Kitagawa group clearly show that the crystal size is a critical factor in structural transitions upon adsorption. [START_REF] Sakata | Shape-Memory Nanopores Induced in Coordination Frameworks by Crystal Downsizing[END_REF] Thus, it is crucial to understand the dynamics of structural transitions at the crystal scale, both from the fundamental point of view and for practical applications of these materials.

By relying upon the fundamental concepts of adsorption stress and osmotic ensemble outlined above, we proposed the first mechanical approach to the adsorption-induced structural transitions, the so-called stress model. [START_REF] Neimark | Stress-Based Model for the Breathing of Metal-Organic Frameworks[END_REF] We suggested that adsorption-induced stress exerted on the material acts as a stimulus that triggers structural transitions of the framework, which occur when the stress reaches a certain critical threshold. Based on the stress model, we provided a quantitative description of the double breathing transitions in MIL-53(Al) upon CO2, Xe, and CH4 adsorption. [START_REF] Boutin | The Behavior of Flexible MIL-53(Al) upon CH4 and CO2 Adsorption[END_REF] Shortly after that, Beurroies et al. [START_REF] Beurroies | Using Pressure to Provoke the Structural Transition of Metal-Organic Frameworks[END_REF] demonstrated that the structural transitions in MIL-53(Cr) could be also set off by uniform mechanical compression of the powder immersed in liquid mercury, as shown in Figure 2. We used this data and as well as the adsorption data and in situ X-ray diffraction patterns reported for CO2 adsorption on MIL-53(Cr) to show that both adsorption and mechanical compression mechanisms of framework transformations could be described within a unified thermodynamic model based on the minimization of the osmotic thermodynamic potential. [START_REF] Neimark | Structural Transitions in MIL-53 (Cr): View from Outside and Inside[END_REF] In particular, we noted that the magnitude of the adsorption stress exerted inside the pores by guest molecules, which is required for inducing the breathing transition, corresponded to the magnitude of the external mercury pressure applied from the outside.

The thermodynamic analysis of equilibrium states within the osmotic ensemble and the adsorption stress model provides a general framework for the description of the structural transition (pore opening, gating, breathing) as a path followed by the system along the free energy landscape from one minimum to another over a barrier, which is crossed upon achieving the threshold stress. The equilibrium transition conditions and respective phase diagrams, as well as the energy barrier can be estimated on the level of single cells by using different simulation approaches. [START_REF] Bousquet | Free energy landscapes for the thermodynamic understanding of adsorption-induced deformations and structural transitions in porous materials[END_REF][START_REF] Bousquet | Adsorption induced transitions in soft porous crystals: An osmotic potential approach to multistability and intermediate structures[END_REF][START_REF] Sugiyama | Adsorption-Induced Structural Transition of an Interpenetrated Porous Coordination Polymer: Detailed Exploration of Free Energy Profiles[END_REF] However, this thermodynamic treatment does not yield any information into the local elastic deformation of the material, nor on the pathways and dynamics of structural transformations.

Most of the earlier studies were focused on key scalar properties, some of them isotropic (bulk modulus), some of them specific to key crystallographic axes (Young's modulus, hardness). [START_REF] Tan | Mechanical properties of hybrid inorganic-organic framework materials: establishing fundamental structure-property relationships[END_REF] Recently, Tan et al. [START_REF] Tan | Exceptionally Low Shear Modulus in a Prototypical Imidazole-Based Metal-Organic Framework[END_REF] have demonstrated on the case of ZIF-8 that state-of-the-art experimental techniques and theoretical calculations can be used to compute the full tensor of second-order elastic constants of MOFs, thus fully accounting for the anisotropy of framework elastic properties. Based on either quantum chemistry calculations or classical molecular dynamics, the tensorial analysis of elastic properties reveals a great deal of information about flexible MOFs.

By using ab initio calculations of the elastic tensor, we demonstrated that high anisotropy and distinct directions of anomalously low Young and shear moduli are the key signatures of "breathing" MOFs, [START_REF] Ortiz | Anisotropic Elastic Properties of Flexible Metal-Organic Frameworks: How Soft are Soft Porous Crystals?[END_REF] in particular those of a wine-rack framework topology, such as the MIL-53 family. [START_REF] Ortiz | Metal-organic frameworks with wine-rack motif: What determines their flexibility and elastic properties?[END_REF] The anisotropy can reach a 400:1 ratio between the most rigid and weakest directions.

Comparison between the characteristic 3D maps of elastic moduli for MIL-47 and MOF-5 structures are shown in Figure 3. While the ratio of the elastic moduli in "soft" and "hard" directions is ~100 for flexible MIL-47, it is only ~2 for non-compliant MOF-5. Further, we showed that the application of hydrostatic pressure on ZIF-8 leads to a shear softening and subsequent mechanical instability, which may cause crystal amorphization. [START_REF] Ortiz | Investigating the Pressure-Induced Amorphization of Zeolitic Imidazolate Framework ZIF-8: Mechanical Instability Due to Shear Mode Softening[END_REF] The characteristic dependences of the unit cell parameter and elastic constants on the confining pressure for ZIF-8 are presented in Figure 4. Another factor that should be taken into account while analyzing the adsorption induced deformation is the influence of the adsorbate of the crystal elastic moduli, which was earlier demonstrated by Coasne et al. drawing on the example of silicalite zeolites. [START_REF] Coasne | Enhanced mechanical strength of zeolites by adsorption of guest molecules[END_REF] We established this effect in MOFs in the case of the ZIF-8 framework. [START_REF] Ortiz | Investigating the Pressure-Induced Amorphization of Zeolitic Imidazolate Framework ZIF-8: Mechanical Instability Due to Shear Mode Softening[END_REF] The detailed modeling of the anisotropic elastic response on the level of single cell provides important information about the deformation pathways and helps identify key order parameters of deformation and structural transitions, which define the framework geometry at given thermodynamic conditions. The next step is to link this microscopic information on mechanical properties to a mechanism of transitions at the level of multiple unit cells. This can be done, at least in a first approximation, by using simple geometric models of the unit cell (depending on its crystal class) with one or few variable order parameters, and writing down the generic equations for elastic compatibility conditions.

In the case of MIL-53(Al or Ga), for example, the analysis of the unit cell elastic tensor [START_REF] Ortiz | Anisotropic Elastic Properties of Flexible Metal-Organic Frameworks: How Soft are Soft Porous Crystals?[END_REF] confirmed that the softest mode of deformation is the cell shear perpendicular to the channel direction. As such, the cell deformation is characterized by one primary order parameter related to the volumetric strain, or to the angle of the pore channel rhombus cross-section, which varies from ~79° in lp states to ~40° in np states (Figure 5, left). In order to couple the deformations of the cells within the 3D framework, we invoked the general theory of elasticity and showed that due to the Saint-Venant elastic compatibility conditions, the structural transition between the lp and np phases happens collectively by shearing of the 2D layer of unit cells, as schematized in Figure 5, right. [START_REF] Triguero | Mechanism of Breathing Transitions in Metal-Organic Frameworks[END_REF] Based on this conclusion, we built the simplest possible, or minimalistic physical model of breathing transitions at the scale of the crystal, including only the most essential elements: host elastic terms (parabolic local term reflecting linear elasticity regime, and elastic coupling between neighboring cells), free energy difference and free energy barrier between the lp and np phases, as well as host-guest coupling in the form of adsorption energy and adsorption stress. [START_REF] Triguero | Understanding adsorption-induced structural transitions in metal-organic frameworks: From the unit cell to the crystal[END_REF] Monte Carlo simulation of coupled adsorption and deformation on the crystal level performed with this minimalistic model allowed us to reveal the origin of the hysteresis arising in adsorption-induced structural transitions, and to gain an insight into the effect of crystal size on macroscopic behavior. We showed a possibility of the phase coexistence within one crystal in the transition region and the dynamic nature of this coexistence. In particular, we found that depending on the system parameters the lp-np transition may occur either reversibly or with a prominent hysteresis, and the cell layers of lp and np phases may be distributed in the crystal either randomly or forming just a few clusters, which rapidly grow in time and coalesce, see Figure 6. Work is now going on to extend this simple qualitative model into a quantitative lattice-based modeling method, by using as input the mechanical and adsorption properties at the microscopic scale, determined either from experiments or from atomistic level molecular simulation (adsorption energy, adsorption strain, elastic moduli). This kind of multiscale approach is fundamental to the understanding, bridging the cell level models and macroscopic dynamics of the crystal structure transformation.

The concepts of the osmotic thermodynamic ensemble and the adsorption stress have found already multiple applications in modeling adsorption deformation phenomena, some of which follow directly the models outlined above and extend them, [START_REF] Wang | Modulating Metal-Organic Frameworks To Breathe: A Postsynthetic Covalent Modification Approach[END_REF][START_REF] Henke | Directing the Breathing Behavior of Pillared-Layered Metal-Organic Frameworks via a Systematic Library of Functionalized Linkers Bearing Flexible Substituents[END_REF] and some of which offer similar or alternative models based on the same basic concepts. In particular, Miyahara and co-workers in a series of papers [START_REF] Sugiyama | Adsorption-Induced Structural Transition of an Interpenetrated Porous Coordination Polymer: Detailed Exploration of Free Energy Profiles[END_REF][START_REF] Watanabe | Free energy analysis for adsorption-induced lattice transition of flexible coordination framework[END_REF][START_REF] Kitaura | Porous Coordination-Polymer Crystals with Gated Channels Specific for Supercritical Gases[END_REF] used free energy calculation methods to study the gate opening behavior in interpenetrated frameworks of "jungle gym" of cubic lattice structure (model systems to interpenetrated and interdigitated coordination polymers with cubic lattice [START_REF] Numaguchi | Simulation study for adsorption-induced structural transition in stacked-layer porous coordination polymers: Equilibrium and hysteretic adsorption behaviors[END_REF] ), and the gate opening and further swelling in stacked-layer pillared structures (which represent Elastic Layerstructured Metal organic-frameworks, or ELMs [START_REF] Kondo | Novel Expansion/Shrinkage Modulation of 2D Layered MOF Triggered by Clathrate Formation with CO2 Molecules[END_REF] ). Pera-Titus and Farrusseng proposed a formulation of thermodynamic isotherms for describing S-shaped adsorption curves in the case of adsorption-induced structural transitions, as well as a way of estimating phase transition energies. [START_REF] Pera-Titus | Guest-Induced Gate Opening and Breathing Phenomena in Soft Porous Crystals: Building Thermodynamically Consistent Isotherms[END_REF] The group of Maurin has used hybrid Molecular Dynamics/Monte Carlo simulations in the osmotic ensemble to study the breathing of materials of the MIL-53 and MIL-47 families, first in presence of adsorbates, [START_REF] Ghoufi | Hybrid Monte Carlo Simulations Com-bined with a Phase Mixture Model to Predict the Structural Transitions of a Porous Metal-Organic Framework Material upon Adsorption of Guest Molecules[END_REF][START_REF] Ghoufi | Physics Behind the Guest-Assisted Structural Transitions of a Porous Metal-Organic Framework Material[END_REF] then with variations in mechanical pressure and temperature. [START_REF] Yot | Large breathing of the MOF MIL-47(V IV ) under mechanical pressure: a joint experimental-modelling exploration[END_REF]49 Very recently, Zhang et al. [START_REF] Zhang | Sorption-Induced Structural Transition of Zeolitic Imidazolate Framework-8: A Hybrid Molecular Simulation Study[END_REF] used a similar methodology to simulate N2 adsorption isotherm in ZIF-8, while accounting for the crucial local framework flexibility of the material. Zang et al. [START_REF] Zang | Osmotic ensemble methods for predicting adsorptioninduced structural transitions in nanoporous materials using molecular simulations[END_REF] applied the osmotic ensemble to study swelling of single-walled aluminosilicate nanotube bundles and cesium montmorillonite. Santander et al. [START_REF] Santander | Simulating Adsorptive Expansion of Zeolites: Application to Biomass-Derived Solutions in Contact with Silicalite[END_REF] performed Monte Carlo simulations in the osmotic ensemble to investigate the expansion of silicalite upon adsorption of biomass-derived solutions. Brochard et al. [START_REF] Brochard | Poromechanics of microporous media[END_REF] attempted to augment the classical Biot poromechanics with the adsorption-induced stress in the case of isotropic deformation. Kowalczyk et al [START_REF] Kowalczyk | Adsorption-induced deformation of microporous carbons: pore size distribution effect[END_REF] determined the pore size dependence of the adsorption stress in microporous carbons. Yang et al [START_REF] Yang | Effects of coal deformation upon CO2 adsorption and sequestration[END_REF][START_REF] Yang | Deformation of coal induced by methane adsorption at geological conditions[END_REF] studied the effects of pore swelling and contraction in coal during carbon dioxide and methane adsorption.

Gor and Neimark [START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids: Macroscopic Approach and Density Functional Theory[END_REF] explained the non-monotonic deformation of mesoporous adsorbents by calculating the adsorption stress resulting from the competition between the disjoining and capillary pressures. Bousquet et al. [START_REF] Bousquet | Adsorption induced transitions in soft porous crystals: An osmotic potential approach to multistability and intermediate structures[END_REF] constructed a general model for analyses of metastability of intermediate structures during adsorption-induced transitions in SPC.

While the coupled thermodynamic and mechanical approach to adsorption-induced deformation and structural transitions in SPC has brought about new physical insight into these phenomena, there are still a number of important open questions that remain to be addressed. In the following, we discuss what we consider as some of the most topical and challenging unresolved problems in this field of research. One of the much-discussed issues is the possibility of phase coexistence within one crystal. Although our minimalistic model, described above, indicates that phase coexistence could occur at certain conditions inside a single crystal, it is not clear under what conditions of time and length scales it may happen in real systems. While in situ X-ray diffraction data [START_REF] Serre | An Explanation for the Very Large Breathing Effect of a Metal-Organic Framework during CO2 Adsorption[END_REF] clearly indicates the coexistence of different phases in polycrystalline samples, no experimental evidence that we are aware of can confirm (or refute) that phase coexistence is possible at the single crystal level, and it seems difficult to design a relevant experiment (to give one example, most MOFs are unstable under electron irradiation at SEM conditions). However, such information could be obtained with in silico experiments, using molecular simulations of a system of sufficiently large scale. One of the great upcoming challenges for molecular simulation is the direct modeling of mechanical properties and stability of deformable structures at this scale, ranging from reversible and hysteretic transformations to irreversible formation of dislocations and fractures. Multiscale approaches such as those described in ref. 37 will be crucial for this, allowing to bridge length and time scales, while existing work on multiscale MOF modeling has until now been mostly limited to the creation of force fields from first principles reference data. [START_REF] Tafipolsky | Systematic First Principles Parameterization of Force Fields for Metal-Organic Frameworks using a Genetic Algorithm Approach[END_REF][START_REF] Tafipolsky | First-Principles-Derived Force Field for Copper Paddle-Wheel-Based Metal-Organic Frameworks[END_REF][START_REF] Mcdaniel | Physically Motivated Force Fields for CO 2Adsorption in Zeolitic Imidazolate Frameworks[END_REF] Another series of open questions is spawned by the recent observation of the very high anisotropy of SPC's elastic properties. [START_REF] Ortiz | Anisotropic Elastic Properties of Flexible Metal-Organic Frameworks: How Soft are Soft Porous Crystals?[END_REF] This anisotropy, which reaches up to two orders of magnitude between the stiffest and softest deformation modes, hints that a scalar treatment of stress and strain, as currently done in both thermo-mechanical models and classical poromechanics, [START_REF]Mechanics and Physics of Porous Solids[END_REF][START_REF] Brochard | Poromechanics of microporous media[END_REF] is likely valid only as a first-order approximation. Further extensions of poromechanics models require a more detailed accounting of the system anisotropy, for example by the introduction of additional (secondary) order parameters that would complement the currently used primary order parameter, which is the volumetric strain. Moreover, a connection needs to be established between the microscopic anisotropy of the system and the overall symmetry of the macroscopic sample. During the transition from the microscopic to the macroscopic scale, the anisotropy is averaged out, and this needs to be accounted for within the mechanical model. The proper upscaling is critically important: while all experimental and theoretical studies of SPCs are done on the level of micrometer-sized crystals, the potential practical applications of these materials will necessarily involve composite materials in the form of films or particles, where anisotropic MOF crystals are glued together by a binder into macroscopically isotropic samples. One example of such process would be the pelletization of MOF powder to produce adsorbents or catalysts. In such a composite system, the phase transformations in SPC's single crystals under action of an external stimulus (adsorbing gas or temperature) could be affected, or even suppressed, by particle-particle and particle-binder constraints. This brings about the confining stress acting on the crystal in addition to the adsorption stress. In order to consider deformation and phase transformation in confined crystals, it is necessary to marry the macroscopic poromechanics treatment at the level of the composite particle and the microscopic thermodynamic and mechanical approaches discussed in this Perspective.

It is worth noting in conclusion that the concepts highlighted and the methods described in this Perspective are not limited to SPCs and are fully applicable to other compliant porous solids.

Phenomena of adsorption deformation and structural transitions in SPCs have attracted a lot of recent interest because of the large-scale stimuli-induced deformations, but flexibility is present in all nanoporous materials to a certain extent. Among the topical problems, where thermodynamic and mechanical models could be of great use, are deformation of micro-and mesoporous adsorbents and catalysts under technological conditions; swelling of clays and laminated minerals; shrinkage and swelling of coal and its influence on the secondary methane recovery and carbon dioxide sequestration; mechanical properties of low-k dielectric films; hydrofracturing in shale-gas reservoirs; hydration-dependent mechanical properties of cement and other building materials; porous membranes and electrodes in fuel cells and batteries, etc.

Although some attempts in these directions have been already made, a comprehensive poromechanics of nanostructured adsorbents still remains to be built.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Different categories of dynamic metal-organic frameworks, each illustrated by a

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Unit cell parameter a and elastic constants C11, C12, and C44 of ZIF-8 as a function of

Author Biographies

François-Xavier Coudert is a CNRS researcher at Chimie ParisTech, a chemical engineering college in Paris, France. His research interests are in the area of molecular adsorption in nanoporous materials, with particular focus on developing theoretical models and tools, as well as molecular simulation algorithms, for the description of Soft Porous Crystals. See http://www.chimie-paristech.fr/molsim for further details.

Anne