Symbolic language for the representation of stimulus-induced structural transitions of responsive frameworksThe key element (judgement) of Freges Begriffsschrift is shown in Fig. 2a. A horizontal line is a content stroke leading, for example, to content A. The thick vertical line on the left affirms the content (or judgement) to be a fact. The vertical line connecting B to A is the condition stroke. This symbol states a condition, leading to a logic judgement. B is a condition for A to be observed, or simplified B leads to A, which is precisely termed logic implication. This vertical line termed condition stroke (B is a condition for A) we adopt for our symbolic language in which B represents a stimulus (condition) to change the structure of the framework (Fig. 2b).Fig. 2: Elementary symbols for illustration of switching processes.a Freges original representation of a logical implication, b elementary step representation for a gate opening process, the red sphere represents the molecular stimulus (butane), the shape of the polygons represents the crystal structure of the framework (closed pore phase – cp, narrow pore phase – np, open pore phase – op, large pore phase – lp), c the small vertical stroke represents the negation, it represents a closed valve for the gas, d desolvation from DMF (pink square), the stimulus is the symbol for a pump (evacuation), e gate opening by nitrogen (green sphere) at 77 K, f no gate opening by nitrogen at 298 K, g butane is desorbed by nitrogen purge at 298 K, leading to gate closing, h specification of the guest stimulus relative pressure p/p0, if p/p0 <gate opening pressure, the framework remains closed, i the gate opening pressure is surpassed, leading to gate opening in adsorption, desorption, and gate closing is observed at lower relative pressure, j breathing process along the Xe (violet sphere) adsorption branch, k alternative representation of a breathing process along the adsorption branch for a framework with cubic np phase, l the representation of phase switching from open to narrow pore state when external pressure is applied.In a strict, logical sense of Freges Begriffsschrift, Fig. 2a states more precisely: does not take place, but one of the other three possibilities takes place:
(1)
A is affirmed and B is affirmed
(2)
A is affirmed and B is denied
(3)
A is denied and B is affirmed
(4)
A is denied and B is denied
For a deeper discussion of Freges Begriffschrift, we refer to the original literature and Supplementary Note 1 (Supplementary Fig. 1)49.In our proposed symbolic language, we target the description of the pore opening or structural transition of a porous material (MOF). We align the states and structural transformations of the framework along the vertical line as it is intuitive for chemists (Fig. 2b). We use the condition stroke to depict the consecutive stimuli changing the framework structure. These structural changes, the spatiotemporal evolution of the framework, are of course highly complex17. However, for simplification and rationalization, it is easier to simplify the phenomena considering only two different states of the framework. The latter is justified for many systems showing bistable characteristics in their energy landscape.At the outset, for simplicity, we only consider the gating process, in which the MOF switches from a closed pore state (cp) or narrow pore state (np), here symbolized by a rhombus into an open pore (op) or large pore (lp) structure, symbolized by a square (Fig. 2b). We omit the vertical thick – affirmation stroke, but introduce an arrow, as it is intuitive for chemists, to indicate the direction of the process in time. The arrow replaces the vertical affirmation stroke, reflecting a transformation that is true in the sense that it has been observed by the experimentalists (Supplementary Note 1). The arrow proposed here differs from the ordinary meaning of the arrow in chemistry in the sense that the transformation is always complete to fulfil the requirements of binary logic. All crystals switch if A is affirmed. Instead, the conventional arrow in chemistry is sometimes used to signify also incomplete reactions which would imply a logic contradiction: “A is affirmed” and “A is denied” at the same time.We can line up all structural changes along consecutive arrows as a history time-line for the material. The stimuli causing structural changes B are arranged below the arrow. This structure provides a clear arrangement of transformations and conditions. The condition B responsible for the pore opening (switching) is a molecular stimulus, a gas entering the pore, for example butane, here symbolized by a red sphere (Fig. 2b). Through gate opening, the gas enters the MOF and stays inside. Hence, the square is filled with red color (butane) after butane exposure.This language is intuitive as the guest inclusion and structural transition are immediately understood, even by a non-expert. At the same time, the concept is versatile, as condition B can be replaced by many symbols individually defined by the experimentalist and expanded to other stimuli (i. e. external pressure, light, electric fields etc.). Specifications may be detailed using technical terms. The symbols for the condition B are arranged in Fig. 3.Fig. 3: Explanation of color codes and meaning of symbols used.aThe specification of temperature and pressure is recommended but may be dispensable if invariable or specified elsewhere; tbd = to be defined (specified) by the experimentalist, b in case of liquids p refers to the outer pressure, not to the vapor pressure. cAs a realistic example, the vapor pressure was selected slightly below the saturation pressure.In the following, we introduce the most important elementary processes for switchable MOFs with this logico-symbolistic language:
Butane leads to pore opening (Fig. 2b).
Without butane, there is no pore opening (Fig. 2c).
Vacuum exposure leads to desolvation and pore closing (also termed activation, Fig. 2d).
For this publication, we have summarized all symbols used in the following in Fig. 3, which may be confusing in the beginning. However, it should be pointed out that many experimental studies only use a limited number of gases (stimuli) and hence need to specify only a few symbols, preserving the simplistic character of the symbolic language using only a few colors with noticeable contrast.The columns for temperature (T) and pressure (p) and other important parameters can be added to specify the symbols further in detail, depending on the study and variations of conditions. For many systems, it may also be helpful to specify the relative pressure (p/p0) or a pressure range instead of absolute gas pressure.We further adopt Freges short vertical stroke in the conditional branch to negate, i.e, express B does not take place, in fact blocking the access of butane to the porous material (Fig. 2c). This representation is also rather intuitive, resembling a closed valve similar to a water lock.Figure 2d symbolizes what is often called activation in porous materials science, an inadequate wording, as it contains no information with respect to the experiment or process. Activation of porous solids is more appropriately termed desolvation and is characterized by guest removal from the pores (typically by vacuum or in gas flow) at defined pressure p (or pressure range) and temperature T (or temperature program). Figure 4 demonstrates changes in powder X-ray diffraction (PXRD) patterns and shape of physisorption isotherm of DUT-8(Ni) in the course of phase transition from op to cp (upon desolvation) and adsorption/desorption (cp to op to cp) with the corresponding schematic representation of the processes.Fig. 4: Elementary symbols for illustration.a desolvation, b adsorption/desorption process, pink square – open pore phase (op) filled with DMF, rhombus – closed pore phase (cp), the green sphere represents nitrogen gas, the green square represents op phase filled with adsorbed nitrogen, c PXRD patterns, d nitrogen physisorption isotherm at 77 K of DUT-8(Ni) in the course of phase transition from op to cp (upon desolvation) and adsorption/desorption (cp to op to cp) (reproduced from ref. 34. with permission from the Royal Society of Chemistry).For studies including variation of adsorption temperatures, the temperature of the experiment should be specified next to the symbol (Fig. 2e, f). With this language and expansion of Fig. 3, important experimental observations are intuitively symbolized:
Nitrogen at 77 K leads to pore opening (Figs. 2e-4b, d).
Nitrogen at 298 K does not lead to pore opening (Fig. 2f).
Purging a butane-filled framework with nitrogen at 298 K leads to pore closing (Fig. 2g).
The gate opening pressure is the characteristic pressure at which the MOF opens its pores. It is a characteristic quantity and specific for the interaction of the gas with the porous solid but also characterizes the activation barrier in gate opening processes. We specify the guest pressure in the scheme next to the symbol (Fig. 2h, i). These statements in a logical way reflect: The MOF opens at a particular gate pressure (Fig. 2i at p/p0 = 0.4 for adsorption switching). In this case, the specific information in the table is omitted and directly transferred to the graph. An alternative graphical representation is outlined in the Supplementary Fig. 2 (Supplementary Note 2). This scheme also takes into account the hysteresis, resulting in a different desorption switching pressure (Fig. 2i at p/p0 = 0.1 for desorption switching).In a more general sense, the pressure is recognized as the characteristic switching pressure. This generalization also easily symbolizes the breathing behavior of frameworks. The most archetypical system is MIL-53 ([Al(OH)(bdc)]n, bdc=1,4-benzene dicarboxylate), which shows a consecutive transformation from lp to np to lp phase during successive increasing the stimulus pressure (Fig. 2j). Here the data approximately reflect the responsive behavior against xenon at 298 K as reported by Ferey52. In principle, the same symbols may also represent the breathing of DUT-49 leading to NGA, but a crystallographer may find it more intuitive to represent this by square symbols as both forms, lp and np phases are both cubic in this case (Fig. 2k)53. Xe has been used to analyze the transitions as the 129Xe NMR chemical shift is highly sensitive to the pore size and allows to analyze NGA, gate opening, and breathing easily, among multiple other versatile in situ analysis techniques34,53,54.There are also examples of framework’s shrinkage under mechanical pressure55,56. The structural transition from large to narrow pore phase for the MIL-53(Cr), MIL-47(VIV) ([VIV(O)(bdc)]n) induced by the isostatic pressure upon mercury intrusion is shown schematically in Fig. 2l55,56.History dependent responsivityAfter introducing the basic symbols, it is possible to represent more complex processes. In particular, the complex history-dependent responsivity is an important target for the communication of more complex observations. Our symbolic language is ideally suited to represent consecutive processes and transformations. An important processing step is the desolvation in vacuum before the framework is exposed to other gases, as outlined above (Fig. 5a).Fig. 5: Representation of history-dependent responsivity.a Desolvation induced pore closing followed by butane (red sphere) induced pore opening, pink square – open pore phase containing DMF, rhombus – closed pore phase, red square – open pore phase containing butane, b simplified representation for desolvation induced pore closing followed by butane induced pore opening, c representation of two consecutive butane adsorption cycles with aging (unknown intermediates), d proper representation of intermediates altered through aging by adsorption cycling, e representation of multiple consecutive butane adsorption cycles with aging (unknown intermediates), f proper distinction of intermediates altered by aging using numbers, g, h distinction of intermediates by their responsivity.Freges two-dimensional graphs allow to arrange the consecutive conditions below each other, resulting in neatly arranged structures (Fig. 5b). This concept also gives access to more complex consecutive procedures.Figure 5e describes a switchable MOF activated from DMF and repeatedly exposed to butane and vacuum, and then does not open anymore. This phenomenon is a characteristic aging by repeated switching reflecting a more complex sample history57. The equivalent two-dimensional graph is depicted in the supporting information (Supplementary Fig. 3, Supplementary Note 3).This example (Fig. 5e) shows the strength of the logico-symbolic language: all transformation steps can be represented without explicit specification (knowledge) of intermediates. In this graph, the intermediates are not symbolized because it was impossible to analyze all structural aspects in between the cycles57. Such a situation is frequently met if the experimentalist does not have enough quantifiable data to characterize all intermediates in detail. Not knowing the cause of the changes in responsivity, inspiration and imagination is needed to guess the techniques suitable to identify the cause.Moreover, the intrinsic logic may be used to derive simple conclusions on intermediates. For this reason, we simplify the history dependence to only two adsorption cycles (Fig. 5c).From our earlier description we know that the MOF opens with butane and closes again when it is desorbed (Fig. 5f). This leaves us with two statements contradicting each other (Fig. 5g), both statements can only be true if cp(1) ≠ cp(2) (Fig. 5h). The cp framework (2) after surpassing one opening and closing cycle is not identical to the starting cp structure (1). In other words, the material records the history (events, transformations) hidden in the “real structure”, which encompasses domain structure, defects, finite size, etc. in contrast to the idealized defect-free periodic crystal structure). After recognizing these changes, the observer could then indicate the changes in the real structure symbolically, even if one does not know exactly the origin or descriptor characterizing this change (Fig. 5d). For simplicity, we have arbitrarily introduced a broken line here. The broken line could indicate some degree of structural deterioration, implying defect formation, milling, or surface deformation. For DUT-8(Ni), the huge volume change indeed causes cracking (shattering) of larger crystallites, leading to a domain structure57. The domain boundaries generate additional strain in the crystals, changing their adaptive behavior and causing framework stiffening. Hence, in this case, the broken line is (more or less) intuitive. In principle, other symbolizations, such as filling with a checked pattern, would also be intuitive symbols for domain formation. However, we recommend to reserve the interior of the polygon for the guests specification while the lines represent the framework.Framework composition or synthesis conditions dependent responsivityWe can also easily use this language to compare MOFs differing in composition. For example, MOFs often differ with respect to the metal in the nodes of the framework15,58. The following diagram shows a comparison of DUT-8(Ni) and DUT-8(Co), pillared layer MOFs containing Ni2- or Co2-paddle wheel nodes, demonstrating only the Ni-MOF to open when exposed to butane while the Co-MOF remains closed (Fig. 6a, b)59.Fig. 6: Dependence of responsivity related to framework structure in a wider sense (real structure).a Impact of the connecting node and framework composition on the adsorption behavior, rhombus – closed pore phase (green – DUT-8(Ni), violet – DUT-8(Co)), red sphere represents butane, b butane physisorption isotherms at 273 K of DUT-8(Ni) and DUT-8(Co) (reproduced from ref. 59. with permission from the Royal Society of Chemistry), c impact of the desolvation process on the framework state, pink square – open pore phase containing DMF, grey square – open pore phase containing DCM, brown square – open pore phase containing ethanol, orange square – open pore phase containing liquid and supercritical CO2, empty square – guest-free open pore phase, d PXRD patterns of DUT-8(Zn) with different structural response depending on desolvation conditions (reproduced from ref. 62. with permission from the Royal Society of Chemistry, CC BY 3.0 licence).Frequently, the synthesis conditions define whether a framework is switchable or not, but we cannot immediately identify the cause for suppressed switchability. The frameworks differ in structure in a wider sense, but the descriptor characterizing the difference is unknown. These differences may relate to defects, minor variations in chemical composition, surface termination, crystal morphology or a certain grain boundary structure induced by the particular synthesis conditions60, but these differences are not immediately recognizable by the experimentalist due to methodological limitations. In this case, it is important to represent the differences in synthesis conditions as part of the history of the MOF and symbolize differences in the framework by a different line color or structure (Supplementary Fig. 4).Framework desolvation dependent responsivityAnother decisive process affecting switchability is the framework desolvation45,61. The symbols for this process have already been introduced above (Fig. 2d).Recently, we reported the impact of framework desolvation on switchability for the model system DUT-8(Zn) [Zn2(2,6-ndc)2dabco]n and found a highly complex interdependence of particle size and activation conditions62. For simplification, we do not consider the particle size effects here and only address the responsivity of the bulk material. We compare two desolvation conditions, solvent exchange with dichloromethane (DCM) followed by evacuation and solvent exchange with liquid CO2 followed by release of CO2 in the supercritical state, and simplify the observations (Fig. 6c). According to PXRD patterns (Fig. 6d), desolvation under dynamic vacuum leads to the phase transition from op to cp phase, while scCO2 drying results in the framework rigidification62.The symbolic language intuitively transfers the information that the desolvation leads to the pore closing as a consequence of DCM removal in vacuum, while the supercritical drying procedure enables the preservation of the op form after evacuation.Multiple phase formationThe foregoing example also reveals a problem of our bivalent logic language. This scheme cannot easily account for the experimental observation of phase mixtures. In the real experiment, it was observed that certain desolvation procedures (in the case of thermal evacuation of DMF from DUT-8(Ni) and DUT-8(Zn)) lead to a mixture of op and cp phases47,62.In principle, one could symbolize the formation of mixtures, as demonstrated in Fig. 7a. However, such a language would imply the loss of all logic implications as yes and no (op and cp) are true at the same time. Instead, from a logical point of view, the observation of multiple phases implies either the sample to consist of more than one real structure (Fig. 7b), or the desolvation conditions (stimulus) are not identical for all crystals (Fig. 7d). This aspect is corroborated by thermodynamic considerations as the coexistence of op and cp phases in equilibrium is only allowed at a single defined pressure of the guest, while our switching observations (start and end points) are mostly far from equilibrium. Hence, at least one kind of crystals is not in equilibrium, and at least two different species must exist. The deeper underlying reason for this problem is that the samples we have discussed so far are, in reality, ensembles of many crystals, and we have tacitly assumed that all crystals are identical. However, in reality, they differ in the real structure and, consequently, in switching behavior.Fig. 7: The problem of multiple phase formation.a Improper representation, b proper representation of multiple phase formation indicating differences in framework structure of the starting material in a wider sense using line coding, pink square – open pore phase containing DMF, rhombus – closed pore phase, empty square – guest-free open pore phase, c PXRD patterns of DUT-8(Zn) crystals (as-synthesized and desolvated by scCO2 drying) (reproduced from ref. 62. with permission from the Royal Society of Chemistry, CC BY 3.0 licence), d representation of multiple phase formation indicating inhomogeneous desolvation conditions using number codes.The term: real structure difference, is deliberately chosen here to cover a wider meaning and could refer to differences in particle size, defect concentration, or surface deformation, etc30. A frequently assumed reason for multiple phase observation in such an example is a wider particle size distribution, and for the fraction of small particles the contraction is suppressed. The implications are important as logics implies that the non-closing particles are different in origin, and the starting sample contains at least two different entities, differing in size or shape or other structural characteristics (Fig. 7b). The multiple phase formation upon desolvation by scCO2 drying was previously reported in macro-sized crystals of DUT-8(Zn), which is shown in Fig. 7c.Inhomogeneous desolvation conditions (Fig. 7d), on the other hand, are a practical obstacle often overlooked. In the catalysis community, shallow bed activation is known to be crucial for good performance. If a porous system is activated as a powder, multiple gradients in the sample may be generated. The vapor desolvated from crystals at the bottom of a flask passes through all crystals on the top layer, leading to differences in their history. Activation ports of adsorption equipment are primitive, and their heating wires cause hot spots. In such systems, the desolvation temperature is not well defined. Evacuation pressures are frequently not even recorded by the observer.An alternative explanation for multiple phase formation far from equilibrium is that some crystals disintegrate into smaller crystals during activation due to desorption stress. However, from a logical point of view, this explanation is equivalent to the situation depicted in Fig. 7b as these crystals disintegrating and remaining rigid somehow differ in real structure (size, shape, defect concentration, or else) in the starting particle ensemble. In essence, from a puristic logical point of view, the framework symbol introduced here can only represent one individual crystal or an ensemble of identical crystals.Crystal size and morphologyRecently, it has been pointed out that crystal size and also morphology play a key role in affecting switchability30,63,64. Crystal size affects the thermodynamics and kinetics of phase transitions65,66. However, for switchable MOFs, particle size effects already play a role at the submicron level, whereas normal binary phases display pronounced size effects only well below 100 nm. The most prominent effect is the suppression of flexibility below a critical crystal dimension. In other words, small crystals do not show switchability. We propose two alternative schemes for representation (Fig. 8a, b).Fig. 8: Effects of particle size and matrix embedding on responsivity.a Identification of finite size by line coding, pink square – open pore phase containing DMF, rhombus – closed pore phase, dashed square – open pore phase, b representation of crystal sizes by differently sized framework symbols, c PXRD patterns of differently-sized DUT-8(Ni) crystals (reproduced from ref. 67. with permission from the Royal Society of Chemistry), d representation of three different crystal size regimes by size-varying symbols (green square – open pore phase containing nitrogen, green sphere – nitrogen), representation of altered (suppressed) responsivity by e polymer adsorption on the outer surface, f embedding in a matrix (yellow line on rhombus and square, red sphere – butane).Scheme b (Fig. 8b) easily allows to describe more complex phenomena. For example, for DUT-8(Ni), it has been shown that macro-sized crystals are flexible and reopen with N2, while nanocrystals ( < 500 nm) are rigid and do not close upon desolvation (Fig. 8b, c)67,68. However, a peculiar phenomenon was observed that intermediate size crystals close but do not reopen (Fig. 8d). The exact reason for such phenomena is still a matter of investigation, but similar behavior has also been reported for DUT-8(Co)59 and DUT-8(Zn)62. A description of such phenomena in ordinary language requires nested if-clauses that create an aversion in the mind of the reader and are difficult to digest and memorize. Our symbolism instead allows a simple visualization and immediate perception of such complex phenomena.Matrix rigidificationCrystals are never isolated entities69. On earth, they are always connected to a surface or substrate, agglomerated with other crystals, intergrown, or covered by side products or polymer binders used to shape MOFs into granules70.Any matrix or interfacial contact alters the responsivity. A simple observation is the matrix rigidification, a stiffening by surrounding components that suppress the pore opening process71,72. An intuitive representation is to introduce such coating on the crystal as an additional line (Fig. 8e).The reversible breathing between the large pore phase and the narrow pore phase of MIL-53(Al) can be altered by the glass matrix (Fig. 8f)73. The encapsulation of MIL-53(Al) crystals within ZIF-62(Zn) matrix allowed to stabilize the metastable large pore phase at room temperature.Selective adsorptionA highly important application of adsorbents is adsorptive separation19,24,74. Switchable MOFs frequently show guest dependent responsivity75,76. A good example to symbolize the high selectivity observed for DUT-8(Co) in contrast to DUT-8(Ni) is the responsivity towards butane and DCM59. While the Ni-based framework shows a non-selective gate opening for both gases (Fig. 9a, c), the Co-based framework selectively opens its pores only for DCM (Fig. 9b, d).Fig. 9: Responsivity of switchable frameworks.a Non-selective pore opening by butane (red sphere) and DCM (grey sphere) for DUT-8(Ni) (shown by green polygons: rhombus – closed pore phase, red and grey squares – open pore phase), b selective pore opening by DCM (DUT-8(Co) shown by violet) and corresponding c, d physisortion isotherms (reproduced from ref. 59. with permission from the Royal Society of Chemistry), e selective pore opening by CO2 in DUT-8(Ni), orange sphere – CO2, f selective adsorption of CO2 in gas mixture adsorption (1st stimulus: CH4 – green sphere, 2nd stimulus CO2 – orange sphere), g selective adsorption of CO2 in mixture phase adsorption (1st stimulus: CO2, 2nd stimulus: CH4), h DUT-163(Cu) framework contraction triggered by parallel application of light and gas adsorption (butane – red sphere), while framework remains in the open phase (empy square) upon irradiation only, i pore opening of DUT-8(Ni) by methanol (light blue sphere) upon pressure, while in the absense of pressure framework remains in the closed phase upon resolvation in methanol.Another important example is frequently considered for applications in CO2/CH4 separation. CO2 can open certain frameworks at defined p, T variables while CH4 under the same conditions cannot open the framework (Fig. 9e). The underlying reason is the adsorption enthalpy, which is higher in magnitude for CO2.The consequence is that even in mixture gas adsorption, an ultrahigh selectivity for CO2 is observed, as symbolized in Fig. 9f, g. No matter in which sequence the gases are introduced, only CO2 enters the pores, leading to pore opening77. The amount of methane in the adsorbed phase is negligible. Certainly, this is a stark simplification, but it facilitates the rationalization. For a more in depth discussion, it would be important to explicitly denote the partial pressures of the guest in the mixture gas phase, which plays the role of the absolute pressure for the single-phase adsorption assuming ideal solutions.Combination of several stimuliMoreover, the phase transition of materials can be induced by the combination of multiple stimuli (Fig. 9h, i)78,79. For instance, DUT-163, a framework based on the linkage of (E)-9,9’-(diazene-1,2-diylbis(4,1-phenylene))bis(9H-carbazole-3,6-dicarboxylate (E-dacdc) to copper(II) dimers, exhibits structural contraction by combined application of light irradiation and adsorption-stress by gas adsorption, although it remains undeformable by light alone (Fig. 9h)78. Structural transformations also depend on the pressure of the guest, and several MOFs respond only at high guest pressure beyond ambient. The cp phase of DUT-8(Ni) is not responsive to alcohols at ambient pressure80, while the phase switching from cp to op phase in MeOH under hydrostatic pressure (up to 0.72 GPa) was demonstrated (Fig. 9i)79.Illustration of complex dependenciesIn our view, symbolic language is particularly useful for analyzing more complex sample histories. We only give one typical example, a guest-dependent shape memory effect58,65,81,82,83,84. In words: A switchable MOF closes the pores if it is desolvated from DCM, however, after adsorption of ethanol, the desorption of ethanol does not lead to pore closing54. Such a complex behavior is difficult to conceive as a sentence but easy to depict in symbolic language (Fig. 10a).Fig. 10: Complex interdependence and logic judgments.a Representation of solvation induced shape memory effect, resolvation of the framework with ethanol leads to a change in responsivity (rhombus – closed pore phase, empty square – guest-free open pore phase, pink square – open pore phase containing DMF, grey square – open pore phase containing DCM, brown square – open pore phase containing EtOH), b Freges original representation of a chain syllogism, c example for chain syllogism through switchable MOFs by interlinked stimuli (blue square – open pore phase containing water, blue sphere represents water, yellow square – hydrogen, aqua sphere – oxygen), d mapping of Freges original chain syllogism with symbolic language and connected stimuli.Logic syllogism, laws of thought, logic gatesFreges Begriffsschrift was a landmark of modern logics for the expression of judgments of pure thought. Our definitions are probably not rigorous enough to satisfy such high standards. We only briefly illustrate the chances and limitations by looking at one of the simple laws of logic, a chain syllogism.The modern representation of a chain syllogism by higher mathematics is the following:$$(({{{\rm{A}}}} \Rightarrow {{{\rm{B}}}}) \wedge ({{{\rm{B}}}} \Rightarrow {{{\rm{C}}}})) \Rightarrow ({{{\rm{A}}}} \Rightarrow {{{\rm{C}}}})$$
(1)
Freges original two-dimensional representation is shown in Fig. 10b. The problem and difference to our language is that statements A, B, C in Freges logic are not classified, they operate without privilege in a column on the right hand, and a distinction of subject and predicate does not occur. In our consecutive language, the history aspect is emphasized by the arrow, connecting the states of the framework and their representations are lined up horizontally. The condition stroke was, up to this point, reserved for the stimulus only. This classification (state vs. stimulus) is an important difference to Freges formula language but more intuitive for the chemist as it resembles an equation for a chemical reaction as outlined in the beginning.For a chain syllogism, we need to state a conditional relationship between two stimuli (A⇒B). Figure 10c illustrates a realistic example. We state the connection between two stimuli as: hydrogen and oxygen form water vapor at 298 K (A⇒B). We select a responsive framework opening the pores when exposed to water vapor (B⇒C). The logic consequence is that this MOF will also be responsive to a mixture of hydrogen and oxygen and should open the pores (A⇒C).Several MOFs with responsivity towards water have been reported. A more recent example of an eightfold interpenetrated MOF with distinct water responsivity has been reported by Roztocki et al23.This example shows an adequate transfer of Freges original scheme. However, some adaption is required in order to represent A as a condition for B, in this example, the stimuli are conditionally interrelated (A⇒B). However, in principle, it should also be possible to consider other general hypotheses (e.g. all framework nanoparticles are rigid) related to the framework structure as a condition and expand the logic analyses further to more complex laws of thought elaborated in the Begriffsschrift49. This endeavor certainly requires further development in the future to derive logical implications and is beyond the scope of this work.An appealing aspect outlined in the introduction is the identification of framework states with a binary coding (cp=0, op=1). This aspect enables the conceptual construction of logic gates. The example outlined in Fig. 10c–d may serve as an illustration to realize an AND gate. It is reasonable to assume that a switchable gating system will neither open the pore by exposure to hydrogen nor to oxygen at 298 K (Fig. 11a, b). The underlying reason is the high temperature far from the respective boiling points. On the other hand, assuming hydrogen and oxygen form water at 298 K, the resulting water vapor will open the pore system (Fig. 11c)23.Fig. 11: Logic gate realization by the switchable framework.a MOF does not open with oxygen shown as aqua sphere (rhombus – closed pore phase), b MOF does not open with hydrogen (yellow sphere), c MOF only opens if oxygen and hydrogen are both present, because they form water (blue square – open pore phase containing water), d MOF does not open with oxygen, e MOF does not open with hydrogen, f MOF only opens if oxygen and hydrogen are both present because they form water, g realization of a resistive logic AND gate with two input gases (A = H2, B = O2), consecutive changes in relative resistance versus a repeated sequence of input gases in switchable JUK-8/carbon nanoparticle /PTFE/Pt composite, a high resistance (1) is only observed if A and B are on (1) while it is low (0) for any other input (CC BY 4.0 licence)85.Hence, only if hydrogen and oxygen are both present as stimuli, the MOF switches from 0 (cp) to 1 (op), as demonstrated recently (Fig. 11c, g)85. This is exactly the definition of a logic AND gate in which two statements need to be true in order to execute the logic operation85. An alternative representation making use of the negation stroke is also illustrated in Fig. 11d–f. This example illustrates the potential of using switchable MOFs for performing logic operations. Realizing complex computing architectures with switchable MOFs is certainly beyond the scope in the near future. But sensing and the simultaneous recognition of multiple stimuli may be within reach using highly selective switchable MOFs and their logic architectures. In particular, the multitude of functionalized linkers and subtle hydrogen bonding interactions in multivariate MOFs provide a versatile platform for complex recognition patterns. In a sense, the very simplistic example given in Fig. 11 resembles an elementary logic recognition process, namely the fact that two stimuli are present simultaneously, leading to a qualitative structure change, a materialized memory of changes in the environment.