Specific Surface Area from Nitrogen Adsorption Data at 77 K Using the

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Specific Surface Area From Nitrogen Adsorption Data at 77 K Using the Zeta Adsorption Isotherm Seyed Arman Ghaffarizadeh, Seyed Hadi Zandavi, and Charles Albert Ward J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08642 • Publication Date (Web): 26 Sep 2017 Downloaded from http://pubs.acs.org on October 5, 2017

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The Journal of Physical Chemistry

Specific Surface Area from Nitrogen Adsorption Data at 77 K Using the Zeta Adsorption Isotherm Seyed Arman Ghaffarizadeh,†,¶ Seyed Hadi Zandavi,†,‡,¶ and C. A. Ward∗,† †Thermodynamics and Kinetics Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Canada, M5S 3G8 ‡Current Address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, U.S.A. ¶These authors equally contributed to this work. E-mail: [email protected]

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ABSTRACT The determination of the specific surface area (SSA) of solid powders has been a long standing problem in surface science. We use the zeta adsorption isotherm (ZAI) and propose a method for determining the SSA of powders using the N2 adsorption measurements at 77 K. The consistency of the results obtained is demonstrated using two α-alumina samples that have different total surface areas. When the proposed method is applied to convert the amount adsorbed per unit mass to the amount adsorbed per unit area, we show that there is no measurable difference between their adsorption isotherms. Also, we show the proposed method can be applied to six different powders having a variation in their specific surface areas of two orders of magnitude. The corresponding error can be obtained from a single equilibrium adsorption measurement and maximum standard deviation in the mean value of the SSA among six cases is less than 7%.


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INTRODUCTION The standard procedure for determining the specific surface area (SSA) has been based on N2 vapor adsorption measurements, interpreted with the Brunauer-Emmet-Teller (BET) 1 isotherm. Several investigators have pointed out that the BET isotherm does not agree uniformly with isotherm measurements, and the value of SSA inferred depends on the portion of the isotherm that is used to determine the SSA. 2–5 The BET method has been suggested to have an accuracy of ± 20%. 6 In the BET isotherm formulation, the conception was that adsorption takes place in layers. 1,7 As the vapor-phase pressure ratio, xV (≡ P V /Ps ), is increased from zero, the adsorbate is assumed to first form a monolayer, and then with a further increase in xV , the molecules were suggested to adsorb on the monolayer to form a second layer and possibly higher layers. In this procedure, regardless of the adsorbate and adsorbent, the monolayer was assumed to be formed at xV equals to 0.4. 2 Recent work has led to development of the zeta adsorption isotherm (ZAI) that has been shown to successfully describe the amount adsorbed in—to date—a total total of 27 different fluid-solid systems. 8–15 At each xV , the ZAI assumes that the adsorbate consists of clusters with different number of molecules. Although the clustering of the water vapor molecules in an adsorbed phase has been hypothesized, 16–21 there is no isotherm other than the ZAI that predicts the number of cluster types in the adsorbate, aζ : the number of clusters consisting of ζ molecules where ζ can be 0, 1, 2, 3 · · · ζm . The main difference between the ZAI and the other isotherms is the prediction of amount adsorbed throughout the entire xV range including the xV approaching unity. Other isotherms predict an infinite amount to be adsorbed in this limit. 22–24 In contrast, the zeta isotherm



indicates that there is an upper limit to the number of molecules in a cluster, ζm . As a results, a finite amount is predicted to be adsorbed at xV equals to unity. 10,14,15 As shown in this paper, the cluster size distribution obtained from ZAI indicates that the monolayer approximation at any pressure is questionable. N2 adsorbing on Si-100024 at 77 K

(a)

(b) 0.8

0.8

0.6

aζ / Mg

0.6 0.4

0.2

0.4

0.6

Vapor-phase pressure ratio, xV

0.8

a1

xV1

0.4

1.0

(c)

0.8

a2

0.2

0.2 0.0 0.0

1.0

a0 / Mg

1.0

Amount of adsorption, nsv(μmol/mg)

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0.2

a3

xV1

0.4 0.2

a4

0.4 0.6 0.8 1.0 Vapor-phase pressure ratio, xV

0.6

1.2

0.2

0.4 0.6 0.8 1.0 Vapor-phase pressure ratio, xV

1.2

Figure 1: (a) The measured amount of N2 adsorbed on silica (Si-1000) at 77 K is shown as solid dots. 25 The solid line was calculated using eq (7) and the values of the isotherm constants are listed in Table 1. (b) Using these listed values, the cluster distributions were calculated from eq (1). The value of aζ in this figure is the number of clusters in the adsorbate that consists of ζ molecules.(c) The fraction of empty sites calculated from eq (5) approaching zero when the pressure is greater than xV = 0.2.

In this work, we justify using the ZAI in the analysis for determining the SSA. We consider the adsorption isotherm measurements of N2 at 77 K on six different solid powders: titania, 2 magnesia, 2 borosilicate glass, 2 silica (Si-1000), 25 and two different α-alumina samples. 2,26 The zeta isotherm is shown to closely agree with the measurements; with the maximum error of 1.6% among all six cases. Afterwards, using the formulation of cluster size distribution, a method for determining the SSA is proposed. The basis of this method is to determine


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the area by calculating the effective projected area of clusters containing various number of molecules on the solid surface. Using the proposed method, the SSA of samples are determined to a maximum of 7% error. In addition, it is shown that the ZAI constants define material properties.

THEORETICAL METHODS The Zeta Adsorption Isotherm for Vapor Adsorption The zeta adsorption isotherm is based on the assumption that the adsorbate consists of molecular clusters with at most one cluster adsorbed at one of the Mg adsorption sites per unit mass of the adsorbent. Each adsorbed cluster was approximated as a quantum-mechanical harmonic oscillator. The possible energies of an adsorbed cluster are assumed to be those of a quantum mechanical harmonic oscillator, with the oscillating frequencies dependent on the number of the particles in the cluster. The maximum number of molecules in a cluster was denoted ζm . Using these assumptions, the canonical ensemble partition function and the associated Helmholtz function of the adsorbate were constructed. The expression for the chemical potential of the clusters containing ζ molecules, µζ , was formulated and used to obtain an expression for the number of clusters that consists of zeta molecules, aζ , where ζ can be 1, 2, 3, . . . up to a maximum ζm

c(αxV − 1)(αxV )ζ aζ = , Mg αxV (1 + c((αxV )ζm − 1)) − 1

ζ = 1, 2, 3 · · · ζm ,


(1)


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where α and c are two temperature dependent functions that depend on the cluster partition function, qζ . The isothermal constant c is related to the partition function of the molecular clusters containing zeta-molecules, qζ , and to the partition function of those with only one molecule, q1 :

qζ = (q1 )ζ (c)1−ζ ,

ζ = 1, 2, · · · ζm .

(2)

The isothermal constant α is related to the partition function through:

α=

µ(T, Ps ) exp , c kB T

q 1

(3)

where µ(T, Ps ) is the reference chemical potential of the adsorbing fluid at the saturation conditions corresponding to the system temperature, T, and kB is the Boltzmann constant. The isotherm constants α(T ) and c(T ) are determined from the measured adsorption isotherm for a particular solid-vapor system. For the number of empty sites per unit mass of adsorbent one can write

a0 = Mg −

ζm X

aζ .

(4)

ζ=1

After combining eq (1) with eq (4), the fraction of unoccupied sites can be obtained

a0 (αxV − 1) = . Mg αxV (1 + c((αxV )ζm − 1)) − 1


(5)

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Table 1: Zeta isotherm Constants and the SSA of Six Solids Determined from N2 at 77 K Adsorbing on Each Solid, Using Both the ZAI and the BET Isotherms Mg c α ∆(ζm ) xV1 AZAI M ABET s s (10−3 µmol/mg) (%) (m2 /g) (µmol/m2 ) (m2 /g) α-alumina 2 140 3.88 102.33 0.80 1.45 0.116 0.396 9.81 0.37 ± 0.05 ± 52.07 ± 0.015 ± 4.31% ± 0.11 Material

ζm

α-alumina 26 100

74.36 ± 2.5

80.48 0.786 ± 22.74 ± 0.014

1.5

0.119

7.61 ± 4.33%

9.76 ± 0.34

6.77

120

2.62 ± 0.19

253.9 0.832 ± 201.45 ± 0.022

1.1

0.065

0.263 ± 3%

9.96 ± 0.73

0.28

magnesia 2 100

2.47 ± 0.06

270.00 0.881 ± 202.45 ± 0.027

1.2

0.651

0.289 ± 2.8%

8.59 ± 0.22

0.26

borosilicate 130 glass 2

1.20 ± 0.01

32.06 0.916 ± 20.08 ± 0.021

1.6

0.164

0.144 ± 7%

8.31 ± 0.03

0.12

silica 118 25 Si-1000

264.33 ± 4.69

534.86 0.748 1.01 0.054 26.314 ± 42.40 ± 0.007 ± 2.05%

10.04 ± 0.13

18.1-26.2

titania 2



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The total amount adsorbed per unit mass of the adsorbent for a given temperature at a solid-vapor interface, nSV g , can be expressed in terms of the cluster distribution of an adsorbate: aζ

nSV g

=

ζm X

ζaζ ,

(6)

ζ=1

and from eq (1) with eq (6)

nSV g

Mg cαxV [1 − (1 + ζm )(αxV )ζm + ζm (αxV )(1+ζm ) ] . = (1 − αxV )[1 + (c − 1)αxV − c(αxV )(1+ζm ) ]

(7)

We note that this expression is written in terms of per unit mass of adsorbent but it can be changed to per unit area of the adsorbent by introducing the specific surface area, As . The number of adsorption sites per unit area of adsorbent, M , is

M=

Mg . As

(8)

and the amount adsorb per unit area is simply

SV

n

nSV g . = As

(9)

We use the Nonlinear Regression package of Mathematica to determine the values of Mg , c, and α that minimize the difference between the measured isotherm and that calculated from eq (7). The error between measured amount adsorbed and the isotherm predictions is assessed using the parameter ∆(ζm ) that, for a specific value of ζ, is a measure of mean-square 8 ACS Paragon Plus Environment

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difference between the measured amount adsorbed and calculated qP Nm ∆(ζm ) ≡

j=1

V 2 V SV [nSV g,mes (xj ) − ng,cal (xj )] , PNm SV V j=1 ng,mes (xj )

(10)

SV where the measured amounts at xV are nSV g,mes and the calculated amounts are ng,cal . For the

total number of measurements Nm the parameter ζm is treated as the threshold value. The threshold is set to be the value after which by increasing ζ further, the error plot plateaus and does not decrease. 8,9 We consider the adsorption isotherm measurements of N2 at 77 K on six solids. The adsorption measurements are used with eq (7) and eq (10) to determine the ZAI constants that are listed Table 1. Having the isotherm constants, the ZAI expression is used to recalculate the amount adsorbed, and the calculated errors are listed in Table 1. Among six cases, the maximum error between the calculations and the measurements, ∆(ζm ), is less than 1.6%. An example of the procedure is shown in Figure 1a for the measured data of N2 adsorbing at 77 K on silica (Si-1000) reported by Jaroniec et al. 25 The solid line shown in the figure was calculated using the described procedure. As seen, the calculation using the ZAI agrees closely with the data throughout entire pressure range. In this case, the error was 1.01% . Using the same information as that used to obtain the results shown in Figure 1a, the fraction of molecular clusters in the adsorbate that have ζ-molecules, aζ /M , and the fraction of the empty site, a0 /M , may be calculated from eqs (1) and (5), respectively. The cluster size distribution results for this case is shown in Figure 1b. The results indicate that for xV near zero, the adsorbate consists primarily of single molecule cluster, but some twomolecule clusters are also present. As the xV is increased, the adsorption sites are occupied 9 ACS Paragon Plus Environment


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by molecular clusters with progressively larger number of molecules. Simultaneously, by increasing xV in the equilibrium range, the fraction of unoccupied sites decreases continuously and approaches zero for xV greater than 0.2, as illustrated in Figure 1c. This figure shows that at the pressure where most of the sites are occupied by single molecule clusters, xV1 , 4.1% of the adsorption sites are still empty. Similar results are found for six other solids.

Specific Surface Area Determination The proposed method for determining the SSA is to calculate the effective projected area of each cluster on the surface of the solid. Since the ZAI does not assume the surface to be covered by the same sized molecular clusters, the variation of the surface area of the different cluster types must be considered. For each cluster type, the number of molecules is known, but the molecules can be oriented in different ways; thus, they can have different projected areas on the surface. The SSA would be summation of the projected area of all clusters on the surface of the solid. Thus, there is an upper and a lower bound for the calculated SSA. The minimum surface area, AZAI s,min is calculated when all the clusters of differently sized are assumed to have their molecules aligned vertically, and the maximum surface area, AZAI s,max , is obtained when the cluster molecules are assumed to be in the planar configuration. If the cluster molecules were aligned vertically, the projected area would be that of a single molecule, and the SSA would be

AZAI s,min

=

ζm X

aζ σca ,

ζ=0


(11)

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N2 adsorbing on α-alumina at 77 K

(a) Amount of adsorption, nsv(μmol/mg)

0.30 0.25



α-alumina25



α-alumina2





0.20

  

0.15 0.10            

  

 

   

  









0.05

0.00 0.0

aζ / Mg





0.2









0.4

a1

x1

a2 a3 0.2

0.4

0.6





     



0.8

1.0

a1

xV1 0.4

a2 a3

a4 0.8

1.0

1.2

0.2

(e) 1.0 0.8

0.6

0.6

a0 / Mg

0.8

0.4



N2 adsorbing on α-alumina2 at 77 K

0.2


1.0



0.6

0.4 0.2



0.6

(c) 0.8

0.8

V




N2 adsorbing on α-alumina25 at 77 K

0.6

(d)



aζ / Mg

(b)

a0 / Mg

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


0.4

0.6

a4 0.8

1.0

1.2


0.4 0.2

0.2 0.2

0.4

0.6

0.8

1.0


0.2

1.2

0.4

0.6

0.8

1.0


1.2

Figure 2: (a) The measured amount of N2 adsorbed per unit mass on α-alumina samples at 77 K is shown as solid dots. 2,26 The solid lines were calculated using eq (7) and the values of the adsorption constants listed in Table 1. (b) and (c) For N2 adsorbing on αalumina samples , the number of occupied adsorption sites with single molecule maximizes at xV1 = 0.118± 1% but there is never a monolayer formed. (d) and (e) Empty sites as functions of xV .For both cases, the numbers of empty sites approach zero after xV = 0.4



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where σca is the effective cross-sectional area of a N2 molecule, 0.162 nm2 /molecule. 27 Note ZAI the area of the empty sites is included in the calculation of AZAI s,min . The As,max would be

reached if all of the cluster molecules were in the planar configuration:

AZAI s,max

=

ζm X

aζ (ζσca ).

(12)

ζ=0

The actual SSA would be expected to lie between the two hypothesized areas.

ZAI AZAI < AZAI s,min < As s,max .

(13)

ZAI The mean of AZAI s,min and As,max ± the % deviation for each of the six solids are listed in

Table 1.

DISCUSSION We consider in more detail the SSA of the two α-alumina samples using adsorption data reported by Matejova et al 2 and independently measured by Cejka et al. 26 The open circles and the solid dots shown in Figure 2a indicate the measured isotherms per unit mass: the two samples differ in the amount adsorbed per unit mass by a factor of 17. The solid lines in Figure 2a show the zeta isotherm recalculation obtained by using eq (7) and the isotherm constants listed in Table 1. The cluster distribution in the adsorbate of two α-alumina samples obtained from eq (1) and ZAI constants are shown in Figure 2b and Figure 2c . It can be seen that the two partial pressures at which the fraction of a1 type clusters reach their highest values are less than 1% different. This supports an earlier 12 ACS Paragon Plus Environment

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contention that c, α, ζm are material properties. 8,28 The mean values of the respective SSA’s of the two α-aluminas are listed in Table 1. Using these values, the amounts adsorbed per unit mass shown in Figure 2a were converted to the amount adsorbed per unit area shown in Figure 3a. There is no measurable difference when the amounts adsorbed are expressed as a surface property: the amount adsorbed per unit surface area. In addition, it can be seen that the converted values strongly agree with the amount measured through out the entire pressure range. The prediction of the amounts adsorbed using BET isotherm is shown in Figure 3b. The solid lines were obtained by fitting the isotherm up to a xV of 0.4 which is the standard procedure for BET fitting. 2,26 It can be seen that the isotherms are only consistent with the measured amounts adsorbed up to a xV of 0.4. As indicated using the cluster size distributions shown in Figure 2b and Figure 2c, for the N2 adsorbing on α-alumina samples at xV1 , single molecule clusters occupy more than 82% of the sites, but a progressively smaller amount as xV approaches unity, and other sized clusters become more important. This indicates that there is never a monolayer formed, according to the ZAI.

CONCLUSIONS A method has been proposed for determining the specific surface area of materials from the ZAI and measured N2 adsorption isotherms at 77 K. The constants appearing in the ZAI are determined from isotherm measurements for 0 < xV < 1, and when the recalculated isotherms are compared with measurements for all six solids, the error is 1.6% or less, Ta13 ACS Paragon Plus Environment


N2 adsorbing on α-alumina at 77 K (b) 100

100 α-alumina25

nsv(μmol/m2)

α-alumina2

60



ZAI

40

  

20    

0 0.0

                           

0.2

 

 

 

 

 

  

0.4 0.6 0.8 Vapor-phase pressure ratio, xV

   

  

nsv(μmol/m2)

80

Amount of adsorption,

(a)

Amount of adsorption,

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1.0

80

α-alumina25 α-alumina2

BET

60 40 20 0 0.0

0.2

0.4 0.6 0.8 Vapor-phase pressure ratio, xV

1.0

Figure 3: (a) The converted measured amount of N2 adsorbed per unit area on α-alumina samples at 77 K is shown as solid dots. 2,26 The solid lines represent the converted amount adsorbed per unit mass to per unit area using the SSA determined from ZAI. Note that while in Figure 2a the maximum amount adsorbed is different by a factor of 17, there is no measurable difference between the converted values in this plot (b) The measured amount of N2 adsorbed per unit area on α-alumina samples at 77 K is shown as solid dots. 2,26 The solid lines represent the BET isotherm fitted to the data up to xV equals 0.4.


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ble 1. The ZAI uniquely treats the adsorbate as consisting of molecular clusters and uses the isotherm constants to predict the number of molecular clusters that contain ζ molecules. The proposed method is based on the cluster distribution prediction of the xV1 value at which single- molecule clusters occupy the largest fraction of the adsorption sites. The efficacy of the method is demonstrated using two α-aluminas that have different capacities for adsorption (µmol/mg), but when the SSA of each was used to convert their respective adsorption capacity to the amount adsorbed per unit area, the calculated isotherms were found to be in close agreement with the data for all xV . When the same procedure was applied using the BET procedure the calculated isotherm were in close agreement, but only in a limited xV range, Figure 3b. Also, while the determination of SSA using the BET method depends strongly on the portion of the isotherm chosen for the calculations, the ZAI method takes the measured data for 0 < xV < 1 into the considerations.

NOTES: The authors declare no competing financial interest.

ACKNOWLEDGMENT: The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, and the European Space Agency.

REFERENCES (1) Brunauer, S.; Emmett, P. H.; Teller, E. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 1938, 60, 309–319.



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(2) Matejova, L.; Solcova, O.; Schneider, P. Standard (master) isotherms of alumina, magnesia, titania and controlled-pore glass. Microporous and Mesoporous Mater. 2008, 107, 227–232. (3) Llewellyn, P. L.; Schuth, F.; Grillet, Y.; Rouquerol, F.; Rouquerol, J.; Unger, K. K. Water sorption on mesoporous aluminosilicate MCM-41. Langmuir 1995, 11, 574–577. (4) Kocherbitov, V.; Alfredsson, V. Hydration of MCM-41 studied by sorption calorimetry. J. Phys. Chem. C 2007, 111, 12906–12913. (5) Kruk, M.; Antochshuk, V.; Jaroniec, M.; Sayari, A. New approach to evaluate pore size distributions and surface areas for hydrophobic mesoporous solids. J. Phys. Chem. B 1999, 103, 10670–10678. (6) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewsks, T. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity. Pure Appl. Chem. 1985, 57, 603–619. (7) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: Mineola, NY, 1986; Chapter 7, pp 134,135. (8) Ward, C. A.; Wu, J. Effect of adsorption on the surface tensions of solid-fluid interfaces. J. Phys. Chem. B 2007, 111, 3685–3694. (9) Zandavi, H.; Ward, C. A. Contact angles and surface properties of nanoporous materials. J. Colloid Interface Sci. 2013, 407, 255–264.


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Graphical TOC Entry 1.0 0.8

Empty Sites

a0/Mg

xV1

0.6 0.4 0.2 0.2

0.4

0.6

0.8

1.0

1.2

xV

0.8 0.6

aζ/Mg

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 20

Cluster Size Distributions

0.4

a1

xV1 a2

0.2

a3 0.2

0.4


0.6

a4 0.8

1.0

1.2

xV

Specific Surface Area from Nitrogen Adsorption Data at 77 K Using the

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