Characterizing Structural Complexity in Disordered Carbons: From the

Subscriber access provided by UNIV OF NEWCASTLE

Invited Feature Article

Characterizing Structural Complexity in Disordered Carbons: From the Slit Pore to Atomistic Models Suresh K Bhatia Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03459 • Publication Date (Web): 08 Dec 2016 Downloaded from http://pubs.acs.org on December 12, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Characterizing Structural Complexity in Disordered Carbons: From the Slit Pore to Atomistic Models Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland St. Lucia, QLD 4072, Australia

ABSTRACT The reliable characterization of nanoporous carbons is critical to the design and optimization of their numerous applications; however, the vast majority of carbons in industrial use are highly disordered, with complex structures whose understanding has long challenged researchers. The idealized slit pore model represents the most commonly used approximation to a carbon nanopore; nevertheless, it has been only partially successful in predicting adsorption isotherms, and fails significantly in predicting transport properties, due to its inability to capture structural disorder and its effect on fluid accessibility. Atomistic modelling of the structure has much potential for overcoming this limitation, and among such approaches hybrid reverse Monte Carlo simulation has emerged as the most attractive. This method reconstructs the structure of a carbon based on fitting of its experimentally measured pair distribution function and appropriate properties such as porosity, while minimizing the energy. The method is shown to be best implemented using a multi-stage strategy, with the first stage used to attain a deep minimum of the energy and subsequent stages to refine the structure based on fitting of specific properties. Methods to determine accessibility of gases based on the atomistic structure are outlined, and it is shown that energy barriers are very sensitive to small differences in the sizes of constrictions and pore entries. The ability to accurately predict macroscopic transport coefficients of adsorbates in nanoporous carbons appears the greatest limitation of such models. Overcoming this will require the fitting of properties more sensitive to long range disorder than the currently used pair distribution, and use of a suitable multi-scaling strategy, which is suggested as a future direction for advancing atomistic models. The inclusion of heteroatoms in the structure is also an important area requiring further attention, particularly in the development of computationally efficient force fields incorporating their interactions. * Email: [email protected]

1 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1. INTRODUCTION The characterization of microporous carbons is a problem of considerable importance because of their numerous applications, conventional and emerging, arising from their strong adsorbing properties. Such carbons have long been the workhorse of industrial separation both in particulate form and as membranes; for example, in gas and liquid phase separations and water purification [1], in protective gas masks and filters for a variety of applications, including warfare, mines, and accidental gas release [2], and in hydrocarbon separation and pollution abatement through capture of volatile organic carbons [3]. They also have a multitude of emerging applications in gas separation and storage including CO2 capture [4], CH4 storage for transportation applications [1], low temperature H2 storage for stationary applications [5], and electrochemical energy storage [6]. On another front, the more recently developed mesoporous carbon aerogels are considered attractive prospects as catalyst supports [7]. Perhaps the most exotic applications of carbons are in nanofluidics, where the use of carbon nanotubes (CNTs) in a host of applications, such as nanoscale fluid transport [8], sensors and actuators [9], and aligned nanotube membranes [10] is being examined. With such a large array of applications, it comes as no surprise that the characterization of nanoporous carbons has long attracted the attention of researchers, and continues to do so. The key challenge here is to model the structural features essential to the prediction of process performance, so as to enable reliable process design. While the structural properties of ordered carbons such as carbon nanotubes are well understood and comprehensively characterized, the conventional microporous carbons in current industrial use have highly disordered and heterogeneous structures, whose accurate representation remains a formidable task. It is this latter class of carbons, and their structural characterisation relevant to their applications in adsorption, that forms the subject of this review. Such carbons possess high internal surface area (typically 1000-2000 m2/g) and pore volume (typically 0.5-1 cm3/g), 2 ACS Paragon Plus Environment

Page 2 of 51

Page 3 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

which enable high equilibrium capacity for fluids in their structure, and make them attractive for the above applications. However, knowledge of these characteristics alone is insufficient to predict adsorption behavior. What are the important structural properties that influence adsorption equilibrium and dynamics in the structure? How are these properties characterized, and how can the characterization results be used to predict fluid equilibrium and transport behaviour in carbons? These are the key questions faced by researchers in the area, and define the scope of this feature article. Disordered carbons possess a morphologically complex internal structure, the understanding of which has long been the subject of intense study [11-17]. While their short range structure is turbostratic, comprising basal plane graphene sheets that are out of registry, there is considerable long range disorder indicative of random aggregation of nanoscale graphitic crystallites [11]. In addition, the graphene layers of the turbostratic structure are highly defective, and exhibit a tortuous and topologically complex surface. Figure 1(a) illustrates the structure of a partially gasified coal char [18], while Figure 1(b) the processed image of a carbide derived carbon (DUT-38) after morphological analysis [19], depicting such twisted sheets in a tortuous structure with short range layering. Nanoscale pores of varying width are also evident in the structures, having walls comprising one or more distorted layers, with disorder at larger scales. A characteristic of such disorganized structures is that the inter-layer spacing increases from the value of 0.335 nm in pure graphite, and can have a wide distribution with values up to as much as 0.346 nm.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 51

(b)

(a)

Figure 1. Transmission electron micrograph of (a) partially gasified coal char [18], and (b) carbide derived carbon, DUT-38, after image processing [19]..

Central to all applications of microporous carbons is the infiltration of fluids in their nanoporous structure. For example, fluid molecules are adsorbed in the carbon nanopores in adsorption based separations or membrane separation processes, with the faster permeating species having higher flux. The nanostructure of microporous carbons is critical to their process performance, as the atomistic details profoundly influence the selectivity, equilibrium, and dynamical properties of infiltrating fluids. It is well established that gases adsorb more strongly in narrow pores due to the stronger interaction with the pore walls, arising from overlap of their potential energy fields . This overlap is most prominent in nanoscale pores, typically smaller than 2 nm in width, termed micropores according to the classification adopted by the International Union of Pure and Applied Chemistry (IUPAC). More generally, nanopores are defined by IUPAC as those smaller than 100 nm and encompass both micropores and mesopores and while the overlap effect may be small in mesopores (pores larger than 2 nm) strong adsorptio near the por wall will arise due to the fluid-solid interaction. Similarly, the pore size and the atomistic details of the solid structure have strong influence on the transport rates and permeability of any fluid in the carbon [20-


Page 5 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

22]. Larger pore sizes afford greater freedom of movement to diffusing molecules, which therefore incur less frequent wall collisions, and are subjected to smaller rate of momentum loss due to wall friction, leading to larger transport coefficients [20, 23]. The frictional loss occurring on wall collision is strongly affected by the atomic structure of the wall [24]. When the surface atoms are closely spaced, as in the ideal surfaces of CNTs or perfectly graphitic slit pores, where the C-C interatomic spacing is 0.142 nm, the generally much larger colliding molecules (typically of size larger than 0.3 nm) experience a rather smooth energy landscape. Consequently, nearly specular collision with small frictional momentum loss is achieved, leading to fast transport with very small Maxwell reflection coefficients (i.e.α ~ 0, where α = 0 corresponds to specular reflection, and α = 1 to diffuse reflection) [25-27], as illustrated in Figure 2. However, in disordered carbons the pore surfaces are highly defective and distorted (c.f. Figure 1), and this leads to increase in frictional momentum loss on wall collision, with commensurate reduction in transport coefficients, as theoretically shown by Roth and Mesentseva using molecular dynamics simulations [28].

α∼0

Hin

α∼1

HCC

Figure 2. Reflection from surface of a slit pore with walls comprising ideal graphene sheets, illustrating relatively smooth surface perceived by large molecules, with smaller molecules being more sensitive to surface texture. The twisting and buckling of the graphene sheets comprising the nanostructure also leads to constrictions and bottlenecks at pore entries, which present energy barriers that can only be


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

overcome at sufficiently high temperature [29, 30]. Conversely, this inhibits access of the infiltrating fluid to the pore space that is only approachable through these pore necks, at low temperatures at which the energy barriers cannot be easily surmounted. The increase in accessibility at high temperatures often leads to unusual effects, such as increase in adsorption with increase in temperature, experimentally observed by Maggs for the low temperature adsorption of nitrogen in some coals [31], and more recently by Bae et al., for argon adsorption at near ambient temperature in coals [32]. While long a subject of controversy, such anomalous increase in adsorption with increase in temperature has been shown to be due to increased pore accessibility at higher temperatures, based on molecular simulation as well as transition state theory (TST) studies [29, 30]. It is clear from the above discussion that the understanding of the carbon nanostructure is critical to prediction of process performance, and to the design and optimisation of processes using this material. Nevertheless, representing the atomistic details of a structure such as that in Figure 1 is a complex task, and it has long been common to consider simpler idealised models. The slit pore depicted in Figure 2 represents the most widely used idealisation of a carbon micropore, with the carbon atoms organized in a graphitic lattice, overlooking shortrange nonidealities associated with the disorder. The relevance of this model is readily seen from the micrograph in Figure 1, where slit-like pore spaces are evident. Although this model does not represent the distortions and irregularities of the graphene layers comprising the structure, it effectively captures the key features essential to the physics of adsorption equilibrium in the structure, viz the presence of a pore space of well-defined shape and size to which a suitable model of adsorption can be applied. For modelling the adsorption in the pore space the classical capillary condensation models based on the Kelvin equation in conjunction with an empirical surface adsorption model have been popular; however, they are known to be inaccurate in pores smaller than 10 nm in width as they 6 ACS Paragon Plus Environment

Page 6 of 51

Page 7 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

overlook interaction between the fluid and the solid [33]. An alternative more suited to micropores is the Dubinin Radushkevich continuous pore filling model, and variations based on this have been widely used [34], but are empirical and lack thermodynamic basis. Thermodynamic models based on the Broekhoff de Boer approach [35], and others based on the Gibbs isotherm, have also been proposed for narrow pores in carbons or other materials [36, 37], but have been found inaccurate in pores smaller than about 7 nm [38]. Such limitations on the lower limit of applicability of the classical models arise due to increasing importance of the molecular texture of the adsorbed fluid at nanoscale dimensions, leading to breakdown of the continuum picture. As a result of these drawbacks, the classical models have been largely superseded by molecular modelling-based approaches such as density functional theory (DFT) [33, 38-42] and Monte Carlo simulation based methods [ 43-46], which are readily accessible at modern computing speeds. These techniques predict the isotherm based on the interaction potential energy distribution in the pore space and are grounded in statistical mechanics. They use independently determined fundamental intermolecular interaction parameters, and predict the isotherm for any pore size with greater accuracy than the classical thermodynamics-based approaches, and without introducing arbitrary fitting parameters. Such approaches are now by far the preferred option, and are available as part of the analysis software suite supplied with commercial instruments for measuring adsorption isotherms. The slit pore model has only been partially successful, as it does not capture pore entries and its idealized structure lacks the distortions and pore mouth constrictions necessary to address the issue of accessibility [29, 30]. Atomistic simulations of the structure hold promise of overcoming the shortcomings of the idealized slit pore model, and have received attention in the last two decades for modelling disordered carbons [12-17, 47, 48]. Ideally, fully mimetic quantum simulations are desirable, as they have the potential to predict the structure based on 7 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

knowledge of synthesis condition alone. However, such simulations are computationally still not feasible, due to the difficulty of dynamically modelling the synthesis process from first principles using a sufficiently large system. Further, the time scales accessible in dynamic first principles simulations are of the order of picoseconds, and far smaller than actual synthesis time scales, which are of the order of seconds to hours. In practice, therefore it has been found more feasible to reconstruct the atomistic structure of a given sample of carbon from suitable experimental data related to the structure. The most successful approach for this is based on the reverse Monte Carlo method, which models the atomistic structure by fitting the measured radial distribution function (RDF) of the carbon while introducing suitable constraints to avoid unphysical structures. The RDF is generally measured using scattering methods, such as x-ray and neutron scattering. First proposed by McGreevy and Pusztai for the structure of liquids [49], the method was subsequently adapted to carbons, with constraints introduced to drive the structure towards one with a coordination number of 3 and bond angle of 120°, consistent with those in an idealized graphene sheet. Nevertheless, highly strained unphysical structures with three and four-membered rings still appeared, and hybrid reverse Monte Carlo (HRMC) simulation methods have since been developed, in which the energy is simultaneously minimised [121-16, 48], using a suitable bond order potential for the carbon [50, 51]. Such methods have proved the most successful in predicting gas adsorption isotherms, however they are unable to capture long range energy barriers [52], due to limitation of the simulation box size which at present is about 3-4 nm. Although larger box sizes of about 10 nm, having as many as 60,000 carbon atoms, are computationally feasible, in practice the radial distribution function lacks resolution at distances beyond about 2-3 nm, and cannot capture the degree of long range disorder. Consequently, larger box sizes are not useful and can actually lead to incorrect structures that are not modulated by experimental data. Recent pseudo-mimetic simulation techniques using quench molecular dynamics do use


Page 8 of 51

Page 9 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

larger box sizes, in which the final structure is modulated by the quench rate [53, 54], which can be tuned to provide reasonable match of mean coordination number or any other experimental quantity. Nevertheless, in the absence of a detailed distance-dependant structural target such as the RDF, such forward simulation methods may not capture the correct level of short range ordering, and further developments using this method are needed before it can be considered an improvement over the HRMC method. In the body of this feature article we discuss in somewhat greater detail the issues of accessibility and prediction of transport coefficients based on slit pore characterization, its successes and limitations, as well as the developing field of atomistic simulations of carbon and how it can overcome some of these limitations. Finally, we discuss critical problems with atomistic modelling, while speculating on potential solutions and future directions in the field.

2. SLIT PORE CHARACTERIZATION OF CARBONS The slit pore idealization is by far the most popular method of characterizing disordered carbons; nevertheless, the disorder in nanoporous carbons leads to considerable heterogeneity and non-uniformity of the pore space, necessitating the consideration of a distribution of pore sizes. Thus, one considers a pore size distribution f(Hin), such that the volume of pores per unit mass of solid having slit-pore width Hin lying in the range (Hin, Hin+dHin) is f(Hin)dHin. Here Hin (= H CC − σ C ) is the geometrical slit width, HCC is the physical slit width (center-tocenter distance between opposing pore walls), and σC is the effective carbon atom diameter, as illustrated in Figure 2. A goal of slit-pore characterisation of nanoporous carbons is to determine this pore size distribution (PSD), and gas adsorption forms the most commonly used technique for this purpose. The PSD is determined by fitting the adsorption isotherm of 9 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 51

a small molecule adsorptive, usually N2 or Ar at its boiling point (N2: 77 K, Ar: 87 K) so that pore filling is achieved, or CO2 at 273 K, using a suitable model to generate theoretical isotherms at any pore size. Thus, one fits the predicted overall excess isotherm, Γ ex ( P) , based on the generalized adsorption integral ∞

Γ ex ( P ) = ∫ [ ρˆ ( H in , P) − ρb ( P)] f ( H in )dH in

(1)

0

to the experimental excess adsorption isotherm to determine the pore size distribution f(Hin). Here P is pressure, ρˆ ( H in , P ) represents the theoretical local absolute isotherm, obtained using DFT or molecular simulation or any other model, in a pore of width Hin, and ρb(P) represents the bulk isotherm. This idealisation implicitly assumes the pores surfaces to be infinite in extent, i.e. the lateral dimensions of the pore are infinite, and that the adsorption in all the pores occurs independently, without any correlation between them. The pore size distribution so determined may then be used to predict the adsorption equilibrium of any other species in the carbon using eq (1), and also model other physical processes in the structure. A weakness of the DFT as commonly applied to slit pore characterization is the assumption of infinitely thick pore walls, when in fact most carbons of interest have high surface areas and pore walls having 1-2 graphene layers [55]. This limitation has been overcome by the introduction of a pore wall thickness distribution (PWTD) which is simultaneously fitted along with the PSD in DFT-based characterization [56-58].

2.1. The Problem of Accessibility An important goal of slit pore characterization using a small molecule adsorbate is to enable prediction of the adsorption of other gases, usually at higher than ambient temperatures, using eq (1) in conjunction with the pore size distribution and surface heterogeneity obtained from the characterization. This is successful in many cases, as demonstrated by Nguyen et al. [58] 10 ACS Paragon Plus Environment

Page 11 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

for the high pressure isotherms of several gases in a number of carbons, using their DFT approach including wall thickness heterogeneity [56-58]. Nevertheless, differences in pore accessibility between the reference molecule used in low temperature characterization, and the higher temperature adsorptive, often lead to failure of the approach. Generally, there is also an accessibility difference between Ar at 87 K and N2 at 77 K, leading to differences in characterization results, further complicating the issue [29, 30]. In general, Ar is able to sample smaller pore sizes in comparison to N2, and has better accessibility at narrow pore widths close to its molecular size (Ar: 0.34 nm, N2: 0.36 nm). Difficulty in equilibration of N2 at 77 K has also been reported by other workers [59], and Ar at 87 K is therefore preferable for characterization, due to its better accessibility in comparison to N2. Nevertheless, even Ar has been shown to have poor accessibility in some carbons, for example in coals at 87 K [32], with isotherms at 313 K showing much larger adsorption. One may expect that CO2 at 273 K will access narrower pores than Ar at 87 K, because of much faster equilibration at the higher temperature and the smaller molecular cros-section (0.33 nm) of CO2. Further, the range of pore sizes having significant adsorption is limited to those below 0.9 nm for CO2 at 273 K, because of its high vapor pressure (34.7 bar) at this temperature while adsorption measurements for characterisation are carried out up to 1 bar. Figure 3(a) depicts the CO2-based PSDs of several Ti3SiC2-derived nanoporous carbons (Ti3SiC2-DCs) synthesized at different heating rates [60], showing similar PSDs, and presence of ultra-micropores as small as 0.32 nm in width in all samples. However, micropores smaller than about 0.4 nm in width are detected only in the sample prepared at 15 K/min heating rate, for the Ar-based PSDs, as shown in Figure 3(b), indicating better accessibility of CO2 in these carbons compared to Ar [66]. The 5 Å Ar-based peaks of the 5 K/min and 15 K/min samples are wider than that of the 2 K/min sample, indicative of lower accessibility of the latter. 11 ACS Paragon Plus Environment

Langmuir

0.12

0.25

(a)

2 K/min 5 K/min 15 K/min

(b) 2 K/min 5 K/min 15 K/min

0.20 3 f(Hin) (cm /g/Å)

0.10

f(Hin) (cm3/g/Å)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 51

0.08 0.06 0.04

0.15

0.10

0.05

0.02 0.00

0.00

3

4

5

6

7

8

9

3

4

5

7

Hin (Å)

10

20

30 40 50

70

100

Hin (Å)

Figure 3. Pores size distributions of nanoporous carbons synthesised from Ti3SIC2 at different heating rates, (a) based on CO2 at 273 K, and (b) Ar at 87 K [60]. Accessibility differences between gases, such as above, suggest that there is no single universal characterization method, and the adsorptive used in low temperature adsorption based characterization must be carefully chosen to match the accessibility of the adsorptive whose high temperature adsorption isotherm is to be predicted. As an example, Figure 4(a) compares the 313 K high pressure isotherms of CO2 in the three carbons, predicted using the Ar PSDs in Figure 3(b), with those determined from experiment. Only the highest Araccessibility sample (15 K/min) yields satisfactory agreement, while the Ar-based PSD under-predicts the isotherms for the other two samples consistent with their lower Ar accessibility. On the other hand the Ar-based PSDs successfully predict methane adsorption isotherm at 313 K in these samples, as shown in figure 4(b), illustrating comparable accessibility of Ar at 87 K and the larger molecule CH4 (0.38 nm) at 313 K in these samples.


Page 13 of 51

8

(a)

2 K/min, pred. 2 K/min, exp. 5 K/min, pred. 5 K/min, exp. 15 K/min, pred. 15 K/min, exp.

12 10 8

excess adsorbed quantity (mmol/g)

14


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

6 4 2

(b)

6

4 2 K/min, pred. 2 K/min, exp. 5 K/min, pred. 5 K/min, exp. 15 K/min, pred. 15 K/min, exp.

2

0

0 0

5

10

15

20

0

5

pressure (MPa)

10

15

20

p (MPa)

Figure 4. Comparison of experimental and predicted isotherms for adsorption of (a) CO2, and (b) CH4 in Ti3SiC2-derived carbon at 313 K [60]. Predicted isotherms are based on PSDs from Ar adsorption obtained using nonlocal DFT considering pore wall heterogeneity [58]. Given the sensitivity of the pores size distribution to accessibility, it is not surprising that predictions of adsorption isotherms of larger molecules based on pore size distributions obtained using Ar or other small molecule adsorptive are often inaccurate, due to the lower accessibility of the larger molecule, which cannot pass through narrow constrictions [45, 58]. The energy barriers associated with such constrictions are also considered responsible for the difficulty in obtaining equilibrium for N2 at 77 K in carbons [29, 30] by imposing diffusional limitations, and the reason why characterization by N2 adsorption has been found unreliable for carbons [59]. Indeed, in his observations of anomalous increase of adsorption of N2 with increase in temperature, Maggs reported the capacity to be dynamic, indicative of diffusional limitations [31]. Figure 5 compares experimental isotherms of CO2 on BPL and PCB activated carbons with those predicted using PSDs based on Ar as well as N2 adsorption [30]. It is evident that the predictions using the Ar-based PSD are in good agreement with the experimental isotherm for both the BPL and PCB carbons (albeit with some very small overprediction at high pressures), indicating that Ar at 87 K and CO2 at the higher temperatures are accessing nearly the same porosity in these carbons. On the other hand the N2 PSD significantly under-predicts the isotherms in BPL carbon, indicating reduced accessibility 13 ACS Paragon Plus Environment

Langmuir

compared to CO2 and Ar. In the case of PCB carbon the N2 PSD-based isotherms satisfactorily match the experimental CO2 isotherm, consistent with the smaller difference in accessibility between Ar and N2 reported for this carbon, in comparison to BPL carbon [30]. 8

(a)

10


12 excess adsorbed quantity (mmol/g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 51

273 K

8

323 K

6 4 2 0

(b)

6

373 K

4

480 K

2

0

0

1

2

3

4

5

0

6

2

p (MPa)

4

6

8

10

p (MPa)

Figure 5. Comparison of finite wall thickness DFT model predicted and experimental adsorption isotherms of CO2 in (a) BPL carbon and (b) PCB carbon [30]. Symbols represent experimental isotherms. Solid and dashed lines depict the predicted CO2 isotherms using the structural parameters (PSD and PWTD) determined from Ar adsorption at 87 K. Similarly, dotted and dashed-dotted lines illustrate the predicted CO2 isotherms, using the structural parameters (PSD and PWTD) determined from N2 adsorption at 77 K.

To include such variable accessibility of the pore space in isotherm predictions, an empirical factor Av is introduced into eq (7), following [45, 58] ∞

Γ ex ( P ) = Av ∫ [ ρˆ ( H in , P) − ρb ( P)] f ( H in )dH in

(2)

0

Here the empirical factor Av is an accessibility factor, used to fit predicted and experimental isotherms. Nguyen et al. [58] report values of Av of 0.91 and 0.86 in for CH4 adsorption in BPL and PCB carbons respectively, using their Ar-based PSD and PWTD respectively. That these value of Av are smaller than unity is an indication of lower accessibility of CH4 to the pore space at the higher temperatures of its adsorption, in comparison to the characterization gas Ar at 87 K in these carbons. This is likely due to the larger molecular size of methane, of 0.38 nm, as compared to the size of Ar of 0.34 nm. As a result of this methane can face larger energy barriers at constrictions and pore entries, that are close to or narrower than its size, but


Page 15 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

which can allow Ar to pass through more readily. Such an accessibility factor was first proposed by Davies and Seaton [45], who found that 10-40% of the pore space in BPL carbon was inaccessible to ethane, but their results suggest that this varies with temperature. Nevertheless, while the different accessibilities of gases pertain to the effects of pore entry energy barriers and are therefore temperature and sample dependent, they do correlate with molecular size for a given sample [58]. The inability of the slit pore model to capture variable accessibility of the pore space without using a fitting parameter reveals an inherent limitation of this idealization, and its lack of reliability when predicting the isotherms of other gases based on characterizations using smaller molecules such as Ar or N2. 2.2

Limitation of the Slit Pore model in Predicting Transport Properties

The above weakness of the slit pore model is even more problematic for transport properties, which may be expected to be very sensitive to the constrictions that impose diffusional limitations on infiltrating molecules, and inhibit accessibility. A second complication is that of surface irregularities and defects existing in the graphene sheets underlying real carbons, but which are absent in the idealized slit pore. Such surface imperfections affect frictional momentum loss on fluid wall collision, since reflection from non-ideal surfaces can be more diffuse than from a defect-free surface. However, the literature is unclear about the relative significance of the two effects: structural constrictions affecting pore mouth barriers and surface irregularities that affect interfacial friction. Comparison of transport coefficients in ideal slit pores and real disordered carbons reveals considerable discrepancy, too large to be rationalized by conventional concepts of tortuosity, commonly used to interpret experimental diffusion coefficients [61, 62]. Figure 6(a) depicts the pore size variation of the theoretical transport diffusivity of CH4 in ideal carbon slit pores


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

with rigid surfaces, based on MD simulations [63], showing it to be of the order of 10-5 m2/s at the pore widths of about 0.5-1 nm typical of disordered carbons. Such large values are comparable to those estimated for ideal carbon nanotubes [27], and are indicative of rapid diffusion with nearly specular reflection and low Maxwell reflection coefficients [25-27]. However, experimental diffusivity values in disordered carbons are smaller by several orders of magnitude; for example values of the order of 10-12-10-8 m2/s have been obtained for the temperature dependent diffusion coefficients of CO2 and CH4 in SiC-derived nanoporous carbon (SiC-DC) [52, 64]. Similar low values have been reported for the diffusivity of H2 in a large number of disordered carbons, based on data compiled from the literature [65], and replotted in Figure 6(b). Except for the highly graphitized carbon black, XC72, SWNH and SWNT, all of which have relatively smooth surfaces, there is a 3-7 orders of magnitude difference between the experimentally measured diffusivities in the disordered carbons and those in Figure 6(a). This difference is far too large to be reconciled by conventional concepts of pore network topology [61-63], and is indicative of failure of the ideal graphitic slit pore model in satisfactorily predicting transport coefficients in carbons. Such failure of the ideal slit pore model has also been recently noted by Coasne et al. [66], in their studies of the structure and dynamics of benzene in ordered and disordered nanoporous carbons, and by Palmer et all in their assessment of this model [67].


Page 16 of 51

Page 17 of 51

10-6 (a)

1 bar 10 bar 40 bar

diffusion coefficient (m 2 /s)

40

diffusion coefficient x 10 6 (m 2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

30

20

10

(b)

10-7 10-8

CG CMS Grafoil PFAC PTMC SWNH SWNT UMC XC72

10-9 10-10 10-11

0 4

8

12

16

20

0

10

20

30

40

50

60

70

-1

pore width (A)

1000/T (K )

Figure 6. (a) Theoretical pore width variation of transport diffusivity of CH4 in carbon slit pores at 298 K, and various pressures, computed using molecular dynamics simulations [63], and (b) temperature dependence of diffusion coefficients of H2 in various carbons compiled from the literature [65]. CG: carbon aerogel, CMS: carbon molecular sieve, PFAC: polyfurfuryl alcohol-derived activated carbon, PTMC: Pt-containing microporous carbon, SWNH: single walled carbon nanohorn, SWNT: single walled carbon nanotube, UMC: commercial ultra-microporous carbon, XC72: graphitized carbon black. While ideal carbon slit pores and carbon nanotubes exhibit a rather smooth surface for most adsorptives, facilitating rapid transport with low values of the Maxwell reflection coefficient of the order of 10-3-10-2 [25-27], very low values of the diffusivity, comparable to those in real disordered carbons, of the order of 10-9-10-8 m2/s have been estimated for CH4 diffusely reflecting in carbon slit pores [20,23] and carbon nanotubes [68]. Nevertheless, this is not necessarily an indication that the surfaces of real carbons are close to being diffusely reflecting. Although surface irregularities are expected to lead to increased deviation of fluid molecule-wall reflections from specularity, molecular dynamics computations of the effect of surface roughness have shown them to reduce the diffusion coefficient of water on graphene by only about 20% [69]. In other work, simulations of the flow of Newtonian fluids on surfaces having random texture in the form of patterns, or atomic heterogeneities through the simultaneous presence of strongly and weakly attractive sites, have shown such irregularities to reduce the transport coefficient by only a small factor of 2-3 and not by orders of magnitude [70]. Consistent with this, simulations of the transport of CH4 in carbon nanotubes considering their flexibility have shown reduction in the diffusion coefficient within a factor 17 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

of 6 [71, 72]. Thus, one expects the surfaces of real carbons to be far from diffusely reflecting (which reduces the transport coefficient by about three orders of magnitude compared to an ideal slit pore), and the contribution of surface defects and irregularities to the transport resistance to be much too small to explain the considerably lower values of experimental transport coefficients in nanoporous carbons in comparison to those in ideal slit pores with rigid surfaces, seen in Figure 6. Indeed, Lim et al. [73, 74] have noted even the diffusely reflecting slit pore model to overestimate the permeability of CH4 and CO2 in carbon molecular sieves by several orders of magnitude. The above arguments suggest that the large discrepancy in transport coefficients between the ideal slit pore and real carbons is predominantly due to the distortions and constrictions that lead to pore mouth barriers and affect accessibility, which cannot be represented by the slit pore model. This view is supported by recent simulations showing as much as two orders of magnitude reduction in the diffusivities of CH4 in carbon nanotubes and slit pores having minor constrictions with openings significantly larger than the size of the CH4 molecule [75]. This inability to capture the structural features that lead to pore entry barriers affecting accessibility as well as fluid transport is a critical drawback of the slit pore model. The developing field of atomistic modelling of carbons holds much promise for overcoming this limitation, and is the subject of the subsequent sections of this article.

3. ATOMISTIC MODELLING OF DISORDERED CARBONS Given the limitations of the slit pore model with regard to prediction of accessibility and transport properties, there has been growing interest in atomistic models of carbons that have the potential to overcome these drawbacks. In one category of these, assumptions are made of the nanoscale units comprising the carbon, such as graphene platelets or nanocrystals [76,


Page 18 of 51

Page 19 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

77], while considering perfect short range ordering within these units. Such models freeze the local degrees of freedom of the carbons within the units, and lack the local heterogeneity inherent to real disordered carbons. Completely atomistic models based on reactive chemistry make no such assumption regarding the short range structure, and form another category that is now receiving increasing attention. In this category, while fully mimetic quantum simulations are still out of reach, Monte Carlo simulations based on mechanical models using generic force fields and an on-lattice initial carbonaceous polymer structure have been reported [78], but the RDF of the final structure derived from the polymeric carbon does not match the experimental RDF satisfactorily. This may be due to the approximate nature of generic force fields, as well as large deviation in starting polymer structure from that of the actual precursor in experiment. While the authors claimed low sensitivity to starting structure in their simulations, this may not necessarily be the case for the significant deviations of the initial structure from that of the actual sample that would be expected, given the uncertainty in the structure of the latter. Similarly, pseudo mimetic models based on quench molecular dynamics simulations [53, 54], while computationally tractable, rely on tuning only the quench rate, and face difficulty in simultaneously reproducing the experimental RDF and adsorption isotherms. Such simulations are initiated by melting the carbon at its known density in a simulation box, and subsequently quenching to room temperature, whence a metastable disordered structure is attained. Figure 7 illustrates the effect of initial state, thermal path and quench rate, on the approach to the final state on a schematic of the free energy landscape [79]. Rapid quench starting from a high temperature melt, or low temperature synthesis from a carbonaceous precursor in a metastable state, leads to a metastable disordered nanoporous carbon (DNC) trapped at a show local minimum of the free energy due to the slow kinetics of the approach to the stable allotrope graphite. On the


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 51

other hand slow quench from high temperature, or synthesis at high temperature at which the kinetics of graphitization is sufficiently rapid, leads to low energy states close to graphite.

Figure 7. Schematic of free energy landscape of carbon, illustrating the effect of synthesis temperature and quench rate on the final state in molecular dynamics simulations [79]. Another technique, the reconstructive HRMC simulation method, has been the most successful to date, and has received much more attention than the above approaches. Consequently, we confine the bulk of our discussion to results obtained using this technique. 3.1 Hybrid Reverse Monte Carlo Simulation 3.1.1 Simulation approach In this method, starting from an initial configuration of carbon atoms in a simulation box at the known experimental density of the carbon, Monte Carlo moves of the carbon atoms are made, through which selected experimentally measured target properties of the structure are fitted, while simultaneously minimizing the energy using a suitable force field [12-16, 48, 80, 81]. Such force fields permit analytical modelling of the effect of local chemistry on the forces between the carbon atoms, allowing tractability of much larger systems compared to first principles calculations.

Among the several force fields available,

Marks’

Environmentally Dependent Interaction Potential (EDIP) [51] has been extensively used in our group, due to its significantly larger cut-off of 0.32 nm compared to the value of 0.2 nm used in the alternatives based on the Reactive Bond Order (REBO) Potential of Brenner and


Page 21 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

co-workers [50]; this may be expected to provide greater fidelity of the short range ordering. In EDIP the potential energy is represented as the sum of two and three body interaction potentials, U2 and U3 respectively, which are dependent not only on the interatomic distance

rij but also on bond angle θijk and a generalized coordination number Zi, i.e.

U i = ∑U 2 ( rij , Z i ) + ∑U 3 ( rij , rik ,θijk , Z i )

(3)

jB)

τdes(B->A)

Temperature

τads(A->B)

τdes(B->A)

(K)

(second)

(second)

(K)

(second)

(second)

273

5.92x10-5

1.86x10-4

200

4.46x107

5.34x107

283

3. 80x10-5

9.54x10-5

253

8.21x103

3.88x103

293

2.53x10-5

5.29x10-5

263

2.48x103

9.90x102

303

1.75x10-5

3.05x10-5

273

8.28x102

2. 80x102

313

1.24x10-5

1.82x10-5

283

3.01x102

8.63x101

323

9.08x10-6

1.12x10-5

323

1.05x101

1.63x100

3.2.3 Influence of pore entry structure and size Critical to the accessibility of any pore body is the structure of its pore mouth, and atomistic models reveal that even very minor differences can lead to significant effects on the crossing times. Figure 13 illustrates this for two synthetic pore mouth models that differ only very marginally in size [30]. Pore mouth model I comprises four concentric circles with each circle containing nine carbon atoms, while model II has similar number of carbon atoms (30 carbon atoms) to that of the pore mouth model I (36 carbon atoms), but comprises only three circles with each circle containing 10 carbon atoms. The top barrier energy value for Ar in pore mouth model I (7.56 KJ/mol) is significantly lower than in pore mouth model II (10.62 KJ/mol), while for CH4 the energy barrier values are very similar (73.6 KJ/mol, 72.3 KJ/mol) in both pore mouth models. As seen in Figure 13 the crossing time values of Ar at 87 K and CH4 at 313 K are 2.9 hours and 34 hours respectively in model I, indicating practically no accessibility problem for Ar at 87 K, but one for CH4 at 313 K. However, reducing the pore mouth size by only a small amount, to 0.31 nm, drastically increases the crossing time of Ar to 200 hrs. (model II), indicating strong sensitivity to even minor changes in size. Such


Page 33 of 51

sensitivity explains the opening up of the nanostructure and large increases in accessible surface area in carbons on activation by small degree of gasification of only 5-10% [86, 87].

0.31 nm

0.32 nm

80

80

pore mouth model II

60

73.6 kJ/mol Ar CH4

40

20

top barrier energy (KJ/mol)

pore mouth model I top barrier energy (KJ/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

7.56 kJ/mol τAr = 2.9 hr.

τCH4 = 34 hr. at 313 K

at 87 K 0

60 Ar CH4

72.3 kJ/mol 40

20

τCH4 = 19 hr.

10.62 kJ/mol τAr = 200 hr.

at 313 K

at 87 K 0 saddle point

saddle point

-20

-20 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

diffusive coordinate (Å)

diffusive coordinate (Å)

Figure 13. Atomistic structure of pore mouths and solid-fluid energy profiles of Ar (solid line) and CH4 (dashed line), for (a) model I and (b) model II [30].

3.3 Modeling the Transport of Gases in Disordered Carbons 3.3.1 Predicting diffusion coefficients in atomistic structures Can atomistic models be reliably used to predict transport coefficients of adsorbates in the nanostructure of disordered carbons? The ability to accurately predict transport properties of guest molecules in carbons requires extreme fidelity of the atomistic model to the structure of the actual carbon; for as discussed above, even minute differences in pore mouth shape or size have very significant influence on the energy barrier experienced by diffusing molecules. Given that the scattering data used for HRMC simulation is always obtained on a bulk sample, the extracted g(r) of the carbon, such as that in Figure 9(b), is a sample average and lacks spatial resolution. It is also short ranged (

Characterizing Structural Complexity in Disordered Carbons: From the

Recommend Documents