Prediction of the Glass Transition Temperatures of Zeolitic Imidazolate

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Chemical and Dynamical Processes in Solution; Polymers, Glasses, and Soft Matter

Prediction of the Glass Transition Temperatures of Zeolitic Imidazolate Glasses through Topological Constraint Theory Yongjian Yang, Collin J. Wilkinson, Kuo-Hao Lee, Karan Doss, Thomas D. Bennett, Yun Kyung Shin, Adri C.T. van Duin, and John C. Mauro J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03348 • Publication Date (Web): 28 Nov 2018 Downloaded from http://pubs.acs.org on December 3, 2018

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The Journal of Physical Chemistry Letters

Prediction of the Glass Transition Temperatures of Zeolitic Imidazolate Glasses through Topological Constraint Theory Yongjian Yang1, Collin J. Wilkinson1, Kuo-Hao Lee1, Karan Doss1, Thomas D. Bennett2, Yun Kyung Shin3, Adri C. T. van Duin1,3 and John C. Mauro1* 1 Department

of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 2 Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, CB3 0FS, UK 3 Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

Corresponding Author: *[email protected] Abstract A topological constraint model is developed to predict the compositional scaling of glass transition temperature (Tg) in a metal-organic framework glass, agZIF-62 [Zn(Im2-xbImx)]. A hierarchy of bond constraints is established using a combination of experimental results and molecular dynamic simulations with ReaxFF. The model can explain the topological origin of Tg as a function of the benzimidazolate concentration with an error of 3.5 K. The model is further extended to account for the effect of 5-methylbenzimidazolate, enabling calculation of a ternary diagram of Tg with a mixture of three organic ligands in an as-yet un-synthesized, hypothetical framework. We show that topological constraint theory is an effective tool for understanding the properties of metalorganic framework glasses. Keywords: zeolitic imidazolate frameworks, glass transition temperature, topological constraint theory

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TOC graphic Predicted Tg for a ternary ZIF glass-forming system: The Tg of [Zn(Im2-x-ybImxmbImy)] can be predicted using a topological constraint model. This model relates the glass transition temperature to the atomic degrees of freedom of the building units in ZIF glass. Other properties, such as porosity, can be optimized while keeping a constant Tg along certain compositional direction.


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Zeolitic imidazolate frameworks (ZIFs) have recently emerged as a new family of glass-forming materials.1–5 Structurally, ZIFs consist of transition metal ions (e.g., Zn2+ and Co2+) linked via organic ligands in an infinite 3D network2,6. As a result of the large variety of structural ‘building blocks’ and network architectures adopted, ZIFs often contain a large amount of functional pores, enabling broad applications in gas separation/storage, catalysis, and drug delivery2,3,6,7. The presence of polymorphism in the ZIF family results in a complex energy landscape. For example, Zn(Im)2 can form multiple zeolitic polymorphs8,9. Comprised of corner-sharing ZnN4 tetrahedra, Zn(Im)2 fulfills Zachariasen’s rules10 for glass formation, proposed in 1932. Recently, new interest has emerged regarding the glass-forming ability of some ZIF compounds, a way to circumvent the obstacles suffered in post-processing of crystalline ZIFs due to their poor physical/mechanical properties6,11,12. While currently the main focus is in the discovery of new ZIFs glasses, understanding the composition dependence of their glass properties, such as glass transition temperature (Tg), has received less attention. The ability to predict such properties would greatly aid the design of ZIF glasses with improved properties. Here, we present a topological constraint model for predicting Tg –– a fundamental glass property which defines both the upper temperature limit for ZIF glasses to be practically used, and the lower limit of liquid formation. Following Angell’s definition13,14, Tg is the temperature at which the supercooled liquid has a viscosity of 1012 Pas. Experimentally, it can be obtained from the temperature where differential scanning calorimetry (DSC) indicates a change in heat capacity, when a sample is scanned at a rate of 10 K/min15. Topological constraint theory (TCT, originally developed by Phillips and Thorpe16) considers the change of atomic degrees of freedom due to bond constraints and proposes that the optimal glass compositions possess atomic degrees of freedom of zero. By considering the connectivity of the glass network, TCT can predict many glass properties including Tg17,18, fragility17,18, chemical durability19, hardness20, and has been widely used 21–24. Our model incorporates a hierarchy of constraints from both the metal and organic ligands. The calculated Tg from the model and the experimental measurement25–27 are in very good agreement,



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with an error less than 4 K. This model can be expanded to predict Tg of other ZIF glasses with different organic ligands. We look at the [Zn(Im2-xbImx)] glass-forming system by considering its fundamental building units: the ZnN4 tetrahedra and Im/bIm ligands, and the constraints able to be imposed (Figure 1). This system is the first ZIF glass with a mixture of two types of organic network formers, and its glass transition temperatures have been reported25–29. ZIF-4 at x=0 was initially found to form a glass11, and ZIF-62 (x=0.25) was later found to be an exceptionally stable glass former25,26. Table 1 details the counting of the bond constraints for each unit in [Zn(Im2-xbImx)], where all constraints are considered to be rigid. As the central quantity in TCT is the network connectivity, hydrogen atoms are not considered since they have only one bond and will never affect the connectivity of the [Zn(Im2-xbImx)] network. In general, all the network-forming species should have a coordination number r  2. Such network-forming species are subject to two types of constraints: (1) two-body constraints (bond stretching, BS), e.g., the  constraint in Figure 1, and (2) three-body constraints (angular bond bending, BB), e.g., the  constraint in Figure 1. According to TCT,16,24 the average number of atomic constraints n for an atom depends on its average coordination number 〈𝑟〉,

𝑛=

〈𝑟〉 2

+(2〈𝑟〉 ―3).

(1)

where r/2 is for BS and 2r-3 is for BB. In ZIFs, there may be over-counting of constraints due to the lack of uniqueness using Equation 1 (explained in the Supporting Information (SI)) which have been accommodated in Table 1. The degrees of freedom per atom, f, is defined as d-n, where d is the dimensionality of the glass network. Depending on the sign of f, the glass network can be either floppy (f >0), isostatic (f=0), or stressed-rigid (f Tx, the constraint becomes floppy (qn → 0) due to the high thermal energy. If we consider only , , and  constraints, their onset temperatures could have the following sequence T < T < T (N-Zn-N angle is quite flexible, even at low temperature30 and the energy barrier of Zn-N bond breaking is fairly high according to Gaillac et al.31). Based on this sequence, the [Zn(Im2-xbImx)] network is stressed-rigid at T < T with a negative f as shown in Figure 2. When the temperature increases and reaches the onset temperature of a certain constraint, e.g., T, f will increase and become positive, as shown by the blue curves. Higher temperature will break stronger constraints ( and ) and make f even larger.

Figure 1. Building blocks of [Zn(Im2-xbImx)] in the box and the mbIm ligand in TIF-4. The atoms are C (gray), N (blue), and Zn (red). Hydrogen atoms are eliminated for they do not affect the network connectivity. Three constraints,  (Zn-N),  (N-Zn-N), and  (N-C2-C3), are exemplified. Table 1. Atomic constraint counting for [Zn(Im2-xbImx)] glass-forming system. BS and BB stand for the radial and angular constraints, respectively. All constraints are considered rigid, corresponding to the situation at low temperature.



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Figure 2. Atomic degrees of freedom (f) based on the bond constraints in the ZIF system shown in Figure 1. The green curve is the adjusted atomic degrees of freedom, see Eq. (4). Using the Adam-Gibbs relation32 and Naumis’s energy landscape analysis33,34, Gupta and Mauro22 have derived the following equation to calculate Tg(x) using a reference composition, xR, 𝑇𝑔(𝑥)

𝑇𝑔(𝑥𝑅) =

𝑓(𝑇𝑔(𝑥𝑅),𝑥𝑅) 𝑓(𝑇𝑔(𝑥),𝑥)

=

𝑑 ― 𝑛(𝑇𝑔(𝑥𝑅),𝑥𝑅) 𝑑 ― 𝑛(𝑇𝑔(𝑥),𝑥)

(3)

.

Here, d=3 for a three-dimensional glass network, n(T, x) is the number of atomic constraint for a composition x at temperature T, and f(T, x) is the atomic degrees of freedom (f = d - n). According 6 ACS Paragon Plus Environment

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to Eq. (3), given Tg of a reference composition, the glass transition temperature of any other glass in the family can be calculated assuming an accurate structural model. The above equation also requires that the average atomic degrees of freedom be positive to yield a meaningful Tg. As can be seen from Figure 2, the atomic degrees of freedom of the system are negative at low temperature; therefore, certain constraints must become floppy in order to make f positive at Tg. Next, we will consider the rigidity of each type of constraints in the [Zn(Im2-xbImx)] system. Following Micoulaut et al.'s35,36 approach and with the bond length/angle varying among different constraint, the relative standard deviation (RSD), i.e., the ratio of the standard deviation of the bond length/angle to its mean, is used. Since the temperature at which decomposition of the organic ligand occurs is far greater than Tg, the BS and BB constraints within the imidazole group and the benzimidazole group are mostly rigid. This is verified using molecular dynamics (MD) simulations of ZIF-62 (detailed in SI), where small RSD (< 0.04) of C-C, C-N bonds and (< 0.05) C-N-C and N-C-N are found for a temperature range varying from 300 K to 1200 K. In contrast, the N-Zn-N angle has a RSD (~0.12-0.19) much larger than the other angular constraints (see Figure S1c), therefore, it can be considered to be floppy, and a RSD of ~ 0.05 may serve as a rough cutoff for rigid constraint. In Figure 3a, the RSD of the Zn-N bond formed between Im/bIm and Zn are compared at different temperatures. Here, it can be clearly seen that the Zn-bIm bond is weaker than the Zn-Im bond, evidenced by a jump at ~500 K of the RSD of Zn-bIm bond. Notice that even though the predicted starting temperature for the Zn-N bIm dissociation is ~500 K in MD, the exact temperature could be higher (see Gaillac et al. in Ref[31]) due to the ZIF ReaxFF force field being trained using small Zn/Im clusters instead of the full ZIF framework. We note several experimental findings: (1) Tg of agZIF-4 (ag stands for glass) is significant lower (300 K) than its melting temperature; (2) Gaillac et al.31 have found the energy barrier of Zn-N dissociation as high as 10.5kBT; and (3) ZIF-4 (cag) transforms into a stable crystalline phase ZIFzni at 673 K upon heating, where Zn atoms are also 4-coordinated. Based on these, it can be safely assumed that the Zn-N(Im) bond-breaking is negligible at Tg of agZIF-4, whereas there may be a small fraction of Zn-N(bIm) bond breaking which serves to decrease the number of constraints. 7 ACS Paragon Plus Environment


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Figure 3b shows the RSD of Zn-N-C1 and Zn-N-C2. The low RSD across the whole temperature range suggests that both constraints are quite rigid. In contrast, the RSD of the C-C-N angle differs between Im and bIm (Figure 3c), where the C-C-N of bIm is floppier than that of Im. Meanwhile, bIm's C-C-C angle is also not as rigid as, e.g., C-N-C (see Figure S1a). Because the C-C and C-N bonds are rigid when we consider the glassy state (see SI), this indicates that there is a fractional constraint difference caused by the relative rigidity of C-C-N bond between Im and bIm and the C-C-C bond of bIm. In addition, such constraint difference can be considered as a constant, since the range of Tg (~40 K) for all the composition explored in Ref[25,27] is relatively small. The above analysis leads us to the following expression for the average number of atomic constraints n of an atom in the [Zn(Im2-xbImx)] system,

𝑛=

2 + 14 ∙ (2 ― 𝑥) + (26 + ∆𝑏𝐼𝑚) ∙ 𝑥

(4)

11 + 4𝑥

where the denominator is the number of total network forming atoms (Zn, N, and C) as a function of the composition and the numerator is the total number of rigid constraints (Zn with 2 rigid BS constraints, Im with 14 constraints and bIm with 26 constraints as shown in Table 1). bIm denotes the constraint change when an Im is substituted by a bIm due to the constraint differences of ZnN, C-C-N, and C-C-C as discussed above. Notice that bIm has been assumed to be independent of temperature for simplification, which is a valid assumption over a narrow range of T.

Figure 3. Relative standard deviation (RSD) of different atomic constraints: (a) Zn-N, (b) Zn-NC, and (c) C-C-N and C-C-C. The inset in (c) shows the atomic index for the C atoms. Combining Eqs. (3) and (4), it follows that 8 ACS Paragon Plus Environment

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[

𝑇𝑔 3 ―

2 + 14 ∙ (2 ― 𝑥) + (26 + ∆𝑏𝐼𝑚) ∙ 𝑥 11 + 4 ∙ 𝑥

]=𝐶

(5)

where C is treated as a fitting parameter related to a reference composition and is constant within a given family of glass22,37. To obtain bIm, we fit Eq. (5) using experimentally measured Tg of [Zn(Im2-xbImx)]25,27 by minimizing the residual sum of squares. The nonlinear curve in Figure 4a is the best fitting (root-mean-square error = 3.5 K) of the experimental data. bIm = -0.5 and C=154.4 K, have been obtained, suggesting that the effective constraint of bIm is 25.5. The equation for calculating Tg is shown in Figure 4a, and the atomic degrees of freedom as a function of x is shown as the green curve in Figure 2. In Figure 4a, we have also included the Tg values of other Zn(Im)2 polymorphs (zec, nog, zni, cag, and GIS) and ZIF-62 from Bennett et al.26 and Zhou28 et al. The thermal histories of these glasses are similar as those in the experimental data used for the model fitting. For agZIF-4 (cag) and agZIF-62, their corresponding crystalline phases have the same topology as [Zn(Im2-xbImx)], and their Tg values are in good agreement with the model. However, the glasses derived from zec, nog, zni and GIS polymorphs of [Zn(Im)2] with different topologies, possess Tg higher/lower than agZIF-4 (cag). This is likely because of different locations of ZIF polymorphs in the potential energy landscape38–40. This has already attracted some attention38 but deserves more study of the ZIF liquid structure, which is beyond the scope of this study. Because TCT has captured the topological reason for the change of Tg, it can be expanded to glasses with different network formers. Here, let us consider TIF-4 [Zn(Im1.5mIm0.5)] as an example. TIF-426,41 has the same topology as ZIF-4 (cag) and ZIF-62, except that Im is partially substituted by mbIm (5-methylbenzimidazolate), similar to bIm but with a -CH3 group on the benzene as seen in Figure 1. The experimental melting point was found to be 740 K, and the resultant glass, agTIF-4 formed upon quenching, possesses a Tg of 616 K26. From the TCT point of view, -CH3 does not affect the connectivity of the network, so it can be ignored in the constraint counting; on the other hand, it may affect the Zn-N bond strength due to its electron donating effect42. The predicted Tg from Eq. (5), ignoring such effect, is 617 K as seen in Figure 4a, which 9 ACS Paragon Plus Environment


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agrees extremely well with the experimental value. Therefore, the net change of the constraint, mbIm, should be also equal to -0.5 when an Im is substituted by a bIm. Given bIm and mbIm, it is instructive to map the composition dependence of Tg for a ternary system like [Zn(Im2-x-ybImxmbImy)] with 0  x and x+y  2, as shown in Figure 4b, where the following equation which combines the effects of both bIm substitution and mbIm substitution (-CH3 is not counted in the total number of atom because it does not change the connectivity of the network but can change the effective number of constraints):

[

𝑇𝑔 3 ―

2 + 14 ∙ (2 ― 𝑥 ― 𝑦) + (26 + ∆𝑏𝐼𝑚) ∙ 𝑥 + (26 + ∆𝑚𝑏𝐼𝑚) ∙ 𝑦 11 + 4 ∙ (𝑥 + 𝑦)

] = 154.4

(6)

where ∆𝑏𝐼𝑚 = ∆𝑚𝑏𝐼𝑚 = ―0.5. The predictive power of TCT can be seen in Figure 4b, where multiple compositions along the red arrow have the same Tg. Other ZIF properties such as mechanical properties43, chemical stabilities44 and porosity42 can then be optimized by selecting different functional groups while keeping Tg unchanged. Note that one can include mixing effect in Eq. (6) by establishing the relation between ∆𝑏𝐼𝑚/∆𝑚𝑏𝐼𝑚 and composition. Meanwhile, the temperature dependence of ∆𝑏𝐼𝑚/∆𝑚𝑏𝐼𝑚 should be also included for accurate prediction when the target Tg is high.


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Figure 4 (a) Comparison of Tg from experiments and the TCT model. Experimental Tg data for [Zn(Im2-xbImx)]25,27 (●), ZIF-zec/nog/zni28 (○), ZIF-4-cag26 (□), Zn(Im)2(GIS)26 (∎), ZIF-6226 (∎) and TIF-426 (∎) are extracted from the literature where similar method has been used to determine Tg. (b) Compositional dependence of Tg for the ag[Zn(Im2-x-ybImxmbImy)] system. Each organic ligand is expressed in percent of the total amount of the organic ligands. In summary, we have developed a topological constraint model to describe the compositional dependence of glass transition temperature in the [Zn(Im2-xbImx)] system. Both experimental and molecular simulation results are considered for different building units in the counting of constraints, which affect Tg. It was found that due to partial bond breakage (Zn-N) and flexible angular bonds (C-C-N and C-C-C), the constraint of bIm is less than the theoretical value considering all of the constraints as fully rigid. This model was further extended to include another type of organic ligand, mbIm, and the composition dependence of Tg for a ternary system [Zn(Im2-xybImxmbImy)]

is obtained by extracting the constraint of mbIm from experimental data. Other

effects, such as temperature dependence of the effective constraint, can be included in the future to enhance the predictability of this model. Our results suggest that TCT can be a valuable tool for understanding thermodynamic properties of ZIF glass systems.

ASSOCIATED CONTENT Supporting Information The supporting information includes: computational details on molecular dynamics; overcounting of constraint in Im and bIm; angular constraint of C-N-C, N-C-N and N-Zn-N. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial or non-financial interests.



ACKNOWLEDGMENT Y.Y. and J.C.M. acknowledge the Institute for CyberScience Advanced CyberInfrastructure (ICSACI) at The Pennsylvania State University for providing computing resources. T.D.B acknowledges the Royal Society for a University Research Fellowship (UF150021).

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Prediction of the Glass Transition Temperatures of Zeolitic Imidazolate

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