Specific Surface Area, Wetting, and Surface Tension of Materials From

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Specific Surface Area, Wetting, and Surface Tension of Materials From N Vapor Adsorption Isotherms 2

Nagarajan Narayanaswamy, and Charles Albert Ward J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b02804 • Publication Date (Web): 08 Jul 2019 Downloaded from pubs.acs.org on July 22, 2019

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The Journal of Physical Chemistry

Specific Surface Area, Wetting, and Surface Tension of Materials From N2 Vapor Adsorption Isotherms Nagarajan Narayanaswamy 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 E-mail: [email protected]

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The Journal of Physical Chemistry 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

ABSTRACT We propose a method for determining the specific surface area, As , of a nanopowder from its measured (mass-specific) N2 vapor adsorption isotherm. The reported crosssectional area of an adsorbed N2 molecule and the (mass-specific) zeta adsorption isotherm are used to calculate As . The (area-specific) zeta adsorption isotherm is then determined using the calculated As . The method is demonstrated by applying it to three nanopowders of each of three materials (α-alumina, carbon, and silica). The values of As and mass-specific adsorption isotherms vary widely. But when As is used to convert the reported adsorption measurements from mass-specific to areaspecific, it is found that they very nearly coincide with the calculated (area-specific) zeta adsorption isotherm. The zeta adsorption isotherm is then used to determine the entropy of the adsorbate. It indicates the wetting phase transition when the substrate is cooled sufficiently. We assume that the surface tension of the solid-vapor interface is transformed to that of the liquid-vapor at wetting. We then combine the zeta adsorption isotherm with the Gibbs adsorption equation. After integrating the result, and applying the wetting condition to evaluate the integration constant, we obtain expressions for the surface tension of the three solids in the absence of adsorption.

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INTRODUCTION It does not seem possible to manufacture nanopowders such that their measured, vaporadsorption isotherms are totally free of heterogeneity effects for all equilibrium values of xV (≡ the vapor phase pressure, P V , divided by the saturation-vapor pressure at the temperature of the isotherm, Ps (T )). This circumstance impedes the determination of surface properties of materials. 1–3 We propose a method for selecting the xV -range from a measured adsorption isotherm of a nanopowder where the heterogeneity effects may be neglected. We use the mass-specific, zeta adsorption isotherm (mZAI), 4–10 and a nonlinear regression analysis to determine the four isotherm constants appearing in the mZAI. When these isotherm constants are combined with the previously reported cross-sectional area of an adsorbed N2 molecule, 11,12 σA , an explicit expression is obtained for the specific surface area, As , of a nanopowder. We demonstrate the method by applying it to each of nine nanopowders that were manufactured from three materials: α−alumina, 13–15 carbon 16–18 and silica. 19–21 Also obtained is the explicit expression for the thermodynamic or area-specific, zeta adsorption isotherm, ZAI. Our objective then was to determine the validity range of the ZAI that was determined from adsorption measurements in the heterogeneity-free range of xV . The values of As for the three samples of each material are found to range over at least two orders of magnitude, and their mass-specific isotherms (mZAI) are three separate functions of xV . When the three ZAI are plotted as a function of xV , they very nearly coincide. The measured adsorption data was converted to the amount adsorbed per unit area using As . The data was then found to lie along a single curve formed by the near-coincidence of the three calculated zeta adsorption isotherms of the three nanopowders of each material. This indicates that a thermodynamic adsorption isotherm is a material property as Gibbs 22 had assumed (but he called it “superficial density”).

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We examine this possibility by supposing the substrate is cooled relative to the vapor, but that the ZAI remains valid at least until the wetting condition is reached. The ZAI is then used to construct the expression for the entropy of the adsorbate, and it indicates that wetting occurs when a substrate is cooled between 2.6 and 2.8 K below that of the vapor. The value depends on the substrate material. We assume that when the wetting transition occurs, the surface tension of the solid-vapor interface is transformed to that of the liquid-vapor interface. We use the ZAI to integrate the Gibbs adsorption equation 22,23 and the proposed definition of wetting 24,25 to evaluate the integration constant. This allows the explicit expression for γ SV to be obtained, and the substrate surface tension in the absence of adsorption, γ S0 , to be calculated. Since γ S0 is a material property, the values found for each material (αalumina, carbon, and silica) sample would be expected to be the same, independently of the specific surface areas of the nanopowders. The values of γ S0 found for each material vary from their mean by 2% or less.

THEORETICAL METHODS Analytical Background of the Zeta Adsorption Isotherm Approach The hypotheses of the zeta adsorption isotherm is that when a vapor adsorbs on a heterogeneityfree solid surface, the adsorbate consists of molecular clusters of several molecules with one cluster adsorbed at one of the Mg adsorption sites per unit substrate mass. The number of molecules in a cluster at an adsorption site, ζ, can be zero or a positive integer: 0, 1, 2, · · · , ζth , where ζth is finite. Each cluster is modeled as a quantum mechanical, harmonic oscillator with a fundamental frequency ω (ζ) that depends on the number of molecules in the cluster. Since the system considered has given values of T, A, and N SV , the canonical ensemble of statistical thermodynamics was used to describe the system statistically. After the partition function was derived with the clusters that consisted of ζ molecules treated as an adsorbed 4


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species, the expression for a chemical potential of a cluster with ζ molecules was obtained, 8 µSV ζ . Note that ζth in this approach is a finite system constant. The system constraints do

not allow an infinite amount to be adsorbed at any xV . Based on these assumptions, the dimensionless fraction of adsorption sites that are empty is given by 26 (αxV − 1) a0g = . Mg αxV [1 + c(αxV )ζth − c − 1]

(1)

As will be seen, the dimensionless, positive isotherm constants α and c are related to the wetting condition and to the effect of adsorption on the surface tension of the solid-vapor interface at wetting. The dimensionless fraction of the sites that are occupied by clusters with ζ molecules is aζg c(αxV − 1)(αxV )ζ , ζ = 1, 2, 3, · · · , ζth . = Mg αxV [1 + c(αxV )ζth − c − 1]

(2)

Note that both a0g /Mg and aζg /Mg depend on the dimensionless isotherm constants, c, α, and ζth . The constants c and α are related to the chemical potential of the adsorbing vapor µSV (T, xV ) through the canonical ensemble partition function of the molecular clusters, qζ , 8,25,26 SV ζ −µ (T, xV ) qζ = c α exp . kB T

(3)

The specific surface area, As , is defined as the area of the substrate per unit mass. And M , is the number of adsorption sites per unit area, and may be expressed as M=

Mg . As

(4)

If we introduce a0 , the number of empty sites per unit area as a0 ≡ a0g /As ,

5


(5)


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then, from eqs (1) and (4) one finds, a0 (αxV − 1) = . M αxV [1 + c(αxV )ζth − c − 1]

(6)

If aζ is defined as the number of clusters per unit area that consists of ζ molecules, then following a similar procedure, aζ /M , can be expressed as: aζ c(αxV − 1)(αxV )ζ = , ζ = 1, 2, 3, · · · , ζth . M αxV [1 + c(αxV )ζth − c − 1]

(7)

The cluster distributions, a0 (xV ) and aζ (xV ), both have apparent singularities at xV equals α−1 , but well defined limits: lim

a0 (xV )/M = (1 + cζth )−1 ,

(8)

lim

aζ (xV )/M = c(1 + cζth )−1 .

(9)

xV →α−1

and

xV →α−1

At xV equal α−1 , the number of clusters that consist of ζ molecules, i.e. clusters of the same type, is predicted have the same concentration in the adsorbate: (M −1 )[a1 (α−1 ) = a2 (α−1 ) = a3 (α−1 ) = · · · = aζth (α−1 )] = c(1 + cζth )−1 .

(10)

It has been previously shown that the canonical ensemble expression for the intensive entropy of the adsorbate, sSV , may be written in terms of the cluster distributions 8,27 ! ζth X sSV (xV ) aζ (xV ) aζ (xV ) , (11) =− ln kB M M ζ=0 where kB is the Boltzmann constant. Below we investigate the entropy-maximum condition and show that this condition indicates a phase transition in the adsorbate. The expression for the amount adsorbed per unit mass of the substrate may be obtained by summing the product ζaζg (xV ) over all nonzero values of ζ. One finds the expression of

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the mZAI in units of the amount adsorbed per unit substrate mass to be V nSV g (x ) =

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

(12)

From the directly measured amount adsorbed per unit substrate mass, a method is proV posed for determining the isotherm constants that appear in nSV g (x ).

The Heterogeneity-free Data Range of Each Nanopowder The measured, mass-specific isotherms of the three α-alumina nanopowders are shown in Figure 1. Those for carbon and silica were similar. Our first objective was to identify the range of xV for which the three mass-specific isotherms of each material were free of micropore filling and capillary condensation effects. First, a nonlinear regression analysis with the mZAI, eq (12), as the model isotherm was used with all of the data reported for each nanopowder. This established the preliminary values of the mZAI constants. When they were used in the mZAI, the color-coded solid lines seen in Figure 1 were obtained. As seen in this figure, there were xV ranges where disagreement was observed. The reason for the disagreement was investigated. As seen in Figures 2a and 2b, in the very low pressure range: 0 ≤ xV ≤ xVµ , the measured adsorption was greater than that predicted with the mZAI. Thus, one possible cause of the disagreement was micropore filling that has been described by others. 18,28–35 Similarly, as seen in Figure 2b, for xVm ≤ xV ≤ 1, the comparison indicates that the amount adsorbed was greater than that predicted by the mZAI. 1–3,16,36 Since the data that was outside the range xVµ ≤ xV ≤ xVm was possibly unreliable, we ask if the data inside the range, i.e., the heterogeneity-free data, could be used to determine reliable values of the mZAI constants: Mg , c, α and ζth . Care must be taken in choosing the value of ζth because a regression analysis will sometimes indicate a value that is larger than the threshold. If an indicated value is the true threshold value, then at xV equal unity,

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N2 vapor adsorbing on α -alumina

0.4

α -alumina 13

0.35

α -alumina 14 α -alumina 15

0.3

nSV (µ mol/mg) 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

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0.25 0.2 0.15 0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

V

Vapor-phase pressure ratio, x

Figure 1: Amount of N2 vapor adsorbed per unit substrate mass on three α-alumina samˇ ples. 13–15 Cejka et al., 13 Jaroniec et al., 14 and Matˇejov´a et al. 15 conducted these experiments at the same liquid nitrogen temperature of 77 K. The three solid lines represent the adsorption isotherms calculated using eq (12) and the isotherm constants Mg , c, α, and ζth , using all the experimental adsorption data.

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N2 vapor adsorption on α-alumina 0.35

α -alumina 13

0.07

α -alumina

0.06

α -alumina 15

0.02

14

0.3

nSV (µ mol/mg) g

0.25

0.05

xV µ

0.04 0.03

nSV g

0.08

nSV (µ mol/mg) 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


0.01 0.85

0.2

0.9

xV

0.95

xV m

0.15 0.1

0.02

xV µ

0.01

xV µ

0

xV m

xV m

0.05 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.65

0.14

0.7

0.75

0.8

0.85

0.9

0.95

1

Vapor-phase pressure ratio, xV


(a) Mass-specific zeta adsorption isotherms (b) Mass-specific zeta adsorption isotherms (color-coded solid lines) and the available data (color-coded solid lines) and the available data in the xV -range where micropore filling is in the xV -range where capillary condensation is thought to be present thought to be present

Figure 2: Amount of N2 vapor adsorbed per unit substrate mass on three α-alumina samples. 13–15 The inset indicates the xV range for α-alumina 15 where the measured data are not approximated as heterogeneity-free. The legend for Figure 2b is the same as shown in Figure 2a. The solid lines represent the amount adsorbed per unit mass determined using data in the heterogeneity-free range and mZAI (eq (12)) with the constants listed in Table 1.

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8,26 nSV g (1 : ζth ) will not change when the value of ζth is increased.

N2 vapor adsorption on α-alumina 0.25

0.015

θ=45 °

α -alumina 14

0.2

Residual(µ mol/mg)

α -alumina 15 ×10 -3

0.15

10

0.1

0.05

8 6

4

6

8

0.05

0.1

0.15

α -alumina 14

0.005

α -alumina 15

0 -0.005 -0.01

1 0 -1 0

-0.02

10 -3 ×10

n SV (nmol/mg) m 0

0.01

-0.015

4

0

α -alumina 13

θ

R (nmol/mg)

α -alumina 13

nSV (nmol/mg) g

Calculated amount, nSV (µ mol/mg) 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

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-0.025

0.2

0.25

5

10

15

nSV (nmol/mg) m 0

0.05

Measured amount, nSV (µ mol/mg) m

0.1

0.15

0.2

0.25

Measured amount, nSV (µ mol/mg) m

(a) Measured and calculated adsorption amount using ZAI

(b) Residuals of measured and calculated adsorption amount

Figure 3: Amount of N2 vapor adsorbed per unit substrate mass on three α-alumina samples. 13–15 The insets on Figures 3a and 3b show respectively the deviations of nSV from g the measured data and the residues for α-alumina. 15 Even though the residuals are negligible, it clearly shows a tendency to deviate at extreme pressure ranges. A second regression analysis was performed using only the heterogeneity-free data. Values of the mZAI constants were determined for each material sample, and are listed in Table 1. For the three samples of α-alumina, we show in Figure 4 a plot of the amount measured to be adsorbed versus the amount calculated to be adsorbed using eq (12) and the values of the isotherm constants listed in Table 1. For xV in the heterogeneity-free range, the difference V SV SV SV of nSV g (x ) with that measured in this range, nexp , is estimated using Er(ng , nexp ) that is

defined as: Nexp SV Er(nSV g , nexp )

≡ kRk2 /

X

V nSV exp (xi ),

(13)

i=1

where kRk2 is the Euclidean norm of the residuals between the measured and calculated V SV V isotherms where the residual, R(xVi ) is the difference between nSV exp (xi ) and ng (xi ). We

include all of the measured points of an experiment where the isotherm may be approximated 10


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0.4

α -alumina 13

0.35


0.3

nSV (µ mol/mg) 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


0.25 0.2

xV m

0.15

xV µ

0.1

xV m

0.05

xV m

0 0

0.2

0.4

0.6

0.8

1

Vapor-phase pressure ratio, xV Figure 4: Amount of N2 vapor adsorbed per unit substrate mass on three α-alumina samples. 13–15 The three solid lines represent the zeta adsorption isotherms calculated using eq (12) and the isotherm constants Mg , c, α, and ζth (Table 1) that were determined from a nonlinear regression analysis of the heterogeneity-free experimental adsorption data. The heterogeneity-free limits, xVµ and xVm , are indicated as vertical dashed lines.

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as heterogeneity-free. The number of data points in this range, Nexp , is listed in Table 1 for SV each nanopowder. The objective of the nonlinear regression analysis was for Er(nSV g , nexp )

to be less than 1.0% between a measured and a calculated isotherm. The maximum value SV of Er(nSV g , nexp ) for the nine nanopowders considered was found to be 0.5%.

The accuracy of the ZAI, is further examined in Figures 3a and 3b. In the former, a plot of the measured amount adsorbed versus the calculated amount adsorbed using the ZAI (eq (12)) and the isotherm constants listed in Table 1, is seen to lie near a 45° line (Figure 3a). In the latter plot, (Figure 3b), the measured adsorption amount versus the residuals for each of the three α-alumina samples are shown. The residuals indicate a tendency to increase in the micropore and capillary condensation regions, but this remains negligible. Thus, in the heterogeneity-free range of xV the ZAI with the constants in Table 1 do describe accurately the amount adsorbed. Our hypothesis is that in the xV ranges outside of the heterogeneity-free range, the amount calculated to be adsorbed is what would have been adsorbed if these ranges were heterogeneity free. Thus we apply the ZAI throughout the equilibrium range of xV . Table 1: Zeta Adsorption Isotherm Constants and Heterogeneity-free Limits for Nitrogen Vapor Adsorbing on α-Alumina, Carbon, or on Silica

Powder

T V (K) Nm

Mg c (nmol/mg)

α-alumina 13 α-alumina 14 α-alumina 15

77 77 77

34 44 19

80.1 26.1 4.2

carbon 16 carbon 17 carbon 18

77 77.4 77

36 32 38

470 106 84

75 57 50

silica 19 silica 20 silica 21

77 77 77.15

44 34 19

288 431 1866

162 120 89

α

ζth

xVµ

xVm a0g (α)/Mg Er

47.8 0.72 37 0.05 0.90 56.7 0.72 37 0.06 0.91 57.7 0.72 37 0.05 0.86

M As (µmol/m2 ) (m2 /g)

0.15% 0.12% 0.12%

0.4% 0.4% 0.3%

10.25 10.25 10.25

7.8 4 0.41

0.73 34 0.08 0.88 0.74 38 0.09 0.86 0.73 41 0.06 0.88

0.07% 0.07% 0.1%

0.3% 0.2% 0.5%

10.25 10.25 10.25

45.9 10.3 8.2

0.70 33 0.01 0.93 0.70 34 0.03 0.85 0.71 32 0.03 0.75

0.07% 0.1% 0.1%

0.3% 0.3% 0.3%

10.25 10.25 10.25

28.4 42.4 182.04

The molecular cross-sectional area of Nitrogen at 77 K is taken as 0.162 nm2 . The value of the error function Er is calculated using eq (13).

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The Specific Surface Area, As , from the Measured Isotherm V A method has been proposed for determining the isotherm constants that appear in nSV g (x ),

eq (12). They may be used to calculate a0g (xV ). One finds it approaches zero asymptotically and is less than 0.15% at xV equal α. Thus at xV (α), essentially each adsorption site is occupied by a N2 cluster. The value of xV (α) for each of the nine nanopowders is given in Table 1. If Mg is the mass-specific number of adsorption sites, then for xV greater than α, each adsorption site would be occupied by a N2 molecule. Although other values have been suggested, when the N2 molecule is in a hexagonal-close packed structure and adsorbed endon, we assume σA has a value of 0.162 nm2 . 11,12 Thus, the specific-surface area of a powder, As , may be expressed as: As = σA Mg NA ,

(14)

where NA is the Avogadro number. Since, by definition, nSV ≡ As nSV , g

(15)

where, nSV (xV ), is the amount adsorbed per unit area of the solid-vapor interface, one finds from eqs (4) and (15) that nSV nSV g = . Mg M

(16)

From eq (12), the ZAI expression for nSV (xV ) is nSV (xV ) =

M cαxV [1 − (1 + ζth )(αxV )ζth + ζth (αxV )1+ζth ] . (1 − αxV )[1 + (c − 1)αxV − c(αxV )1+ζth ]

(17)

The number of adsorption sites per unit interface area may be expressed in terms of σA by

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combining eqs (4) and (14): M = (σA NA )−1 .

(18)

And from eq (18), one finds that for N2 vapor adsorbing on any solid surface, M has the value M = 10.25

µmol . m2

(19)

Note that the proposed method for determining the surface area of a nanopowder leads to the value M having the same value for N2 adsorbing on any substrate.

RESULTS The Thermodynamic Isotherms of α-Alumina, Carbon, and Silica Our objective now is to determine if the isotherms defined by eqs (17) and (19) with the values M, c, α, and ζth listed in Table 1 satisfy the Gibbs criterion for an isotherm. He treated the thermodynamic isotherm as a unique property of a material that, independently of its specific surface area, described the amount adsorbed per unit surface area. Therefore, it could be used as the basis for predicting other properties of a material, such as its surface tension. 22 We illustrate the issue using the three mass-specific isotherms of α-alumina shown in Figure 4, compared with the three thermodynamic isotherms shown in Figure 5. As indicated in Figure 4, the three measured mass-specific adsorption isotherms do not depend uniquely on xV . Each of the material samples has its own dependence. By contrast, independently of the different specific surface areas, for the three samples of α-alumina, the thermodynamic isotherms shown in Figure 5 depend uniquely on xV , and therefore they coincide to form a single curve when plotted together as a function of xV . This coincidence indicates they 14


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50


40

nSV(µ mol/m2 )

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


α -alumina 15 Heterogeneity-free range

30

20

10

xVm

xVµ 0 0

0.2

0.4

0.6

0.8

1

Vapor-phase pressure ratio, xV Figure 5: Amount of N2 vapor adsorbed per unit surface area on each of the three α-alumina samples 13–15 at 77 K. The solid lines represent the adsorption isotherms per unit area, nSV , calculated using eq (17) with the isotherm constants M, c, α, and ζth , (Table 1). The heterogeneity-free experimental data are indicated by the symbols. The average heterogeneity-free limits xVµ ≤ xV ≤ xVm are shown as vertical dashed lines.

15



N2 vapor adsorbing on carbon

50

carbon 16 carbon 17

40

nSV(µ mol/m2 )

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

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carbon 18 Heterogeneity-free range

30

20

10

xVm

xVµ 0 0

0.2

0.4

0.6

0.8

1

V


Figure 6: Amount of N2 vapor adsorbed per unit area on each of the three carbon samples. 16–18 Silvestre et al. have reported data for carbon 17 at 77.4 K and Kruk et al. 16,18 did the investigation for carbon samples 16,18 at 77 K. The solid lines represent the thermodynamic adsorption isotherms calculated using eq (17) and the isotherm constants M, c, α, and ζth (Table 1). The heterogeneity-free experimental data are indicated by the symbols. The average heterogeneity-free limits xVµ ≤ xV ≤ xVm are shown as vertical dashed lines.

16


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N2 vapor adsorbing on silica

50

silica 19 silica 20

40

nSV(µ mol/m2 )

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


silica 21 Heterogeneity-free range

30

20

10

xVm

xVµ 0 0

0.2

0.4

0.6

0.8

1

V


Figure 7: Amount of N2 vapor adsorbed per unit area on three (partially hydroxylated) silica samples. 19–21 Baker and Sing investigated the adsorption on silica 21 at 77.15 K. Nitrogen adsorption data on silica reported by Bhambhani et al 20 and Jaroniec et al. 19 were done at 77 K. The solid lines represent the thermodynamic adsorption isotherms calculated using eq (17) and the isotherm constants M, c, α, and ζth (Table 1). The heterogeneityfree experimental data are indicated by the symbols. The average heterogeneity-free limits, xVµ ≤ xV ≤ xVm , are shown as vertical dashed lines.

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Page 18 of 40

satisfy Gibbs’ criterion for being a thermodynamic adsorption isotherm of α-alumina. The same procedure was followed to determine the thermodynamic isotherms of each of the three samples of carbon and the three samples of silica. Importantly, when plotted together, the three thermodynamic isotherms of each material also coincide (within the error bars) to form a single curve, as shown in Figures 6 and 7. Table 2: Solid Surface Tension, Wetting Condition, and Specific Surface Area for α-Alumina, Carbon, or Silica samples As N2 (m2 /g) vapor, T V (K)

Solid

α-alumina 13

Wetting condition TwS (K)

Sub-cooling (T V − TwS ) (K)

γ S0 (mJ/m2 )

ABET (m2 /g)

δSSA %

7.8

77

74.4

2.6

58.4 ± 0.8%

6.77

14

α-alumina

14

4

77

74.4

2.6

59.5 ± 0.8%

3.56

11

α-alumina

15

0.41

77

74.4

2.6

59.6 ± 0.6%

0.37

10

12

γ S0 = 59.1 ± (RSD) 1.2% carbon 16

45.9

77

74.5

2.5

60.8 ± 0.6%

40.20

carbon

17

10.3

77.4

74.6

2.8

59.7 ± 0.4%

−

carbon

18

8.2

77

74.5

2.5

59.3 ± 1%

6.2

24

γ S0

= 59.9 ± (RSD) 1.4%

silica 19

28.4

77

74.2

2.8

64.5 ± 0.6%

18.1 to 26.2

8 to 36

silica

20

42.4

77

74.2

2.8

63.2 ± 0.6%

38.7

9

silica

21

182.04

77.15

74.3

2.85

61.7 ± 0.6%

164.1

10

γ S0 = 63.1 ± (RSD) 2.1% The calculation of As is based on the assumption that σA (N2 ) has a value 37 of 0.162 nm2 . The solid surface tension in the absence of adsorption is given by γ S0 . The value for the liquid-vapor surface tension, γ LV , of nitrogen 38,39 at 77 K is taken to be 8.9 mJ/m2 . The quantity γ S0 − γ LV is the reduction in γ SV due to adsorption. The relative standard deviation (RSD) for γ S0 is calculated using the mean value of γ S0 . The quantity ABET , is the BET area reported by the reference. The percent difference, δSSA , between As and the BET area, ABET , is determined using eq (20).

The ZAI and the BET Specific Surface Areas For eight of the nine systems that we considered, the BET areas were reported in the respective articles, along with their data. This allowed us to compare the corresponding 18


Page 19 of 40 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


ZAI areas with eight BET areas. In Table 2 both the values of the specific surface areas obtained from the ZAI method, As , and the reported values obtained from the BET method, ABET , are listed using the same data in each case. Also the percent difference, δSSA is defined as: δSSA =

As − ABET , As

(20)

and its values are listed in Table 2 for each of the eight isotherms. The largest percentage difference for α-alumina is 14 % and for carbon it is 24%. For silica, Jaroniec et al. have determined the BET area using different pressure ranges to conduct the BET analysis. Due to this variation in the BET area, the percentage difference with As is 8 to 36 %. 19 Clearly the BET surface area is “dependent” on the data range used to define the “monolayer capacity”. 40 In the ZAI approach, the monolayer capacity concept is replaced by Mg . Its value is determined with a nonlinear regression analysis using all of the equilibrium, heterogeneity-free data, and with nSV as the model isotherm. g Other methods have been proposed with the objective of improving the BET methodology for determining the surface area. These include the t-method, 40,41 the α-plot method, 20,40 and the Inflection-point method. 42,43 Collins et al., 44 have investigated the α-plot and Inflectionpoint methods for silica samples 19,20 and found that the areas were similar to the previously reported BET values.

The ZAI and the BET Isotherms In section 5.1, the value of the specific area determined by the ZAI and BET methods were discussed. For α-alumina, 13 the difference was 14%. But this does not mean that the isotherms are within 14% of one another. Plots of the BET adsorption isotherm per unit area using the BET area and the zeta adsorption isotherm per unit area using As are seen in Figure 8. Note that the two isotherms are in close agreement only up to xV ≤ 0.4. Above this value of xV only the ZAI is in agreement with the data. For values of xV beyond 0.4, it

19



is seen that BET predicts higher adsorbed amounts than the data, and an unphysical infinite amount to be adsorbed when xV reaches unity. The infinite adsorption issue of the BET model has been addressed by other adsorption models, 5,45,46 but to our knowledge, those models have not led to explicit expressions for the specific surface areas.

Nitrogen adsorption on α -alumina at 77 K

40

35

BET Isotherm

35

30

30

25

25

ZAI 20 Isotherm

20

15

n SV ( µ mol/m 2 )

45

2 n SV ( µ mol/m ) BET

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 40

15 10

10

5

5 0

0 0

0.2

0.4

0.6

0.8

1

Vapor-phase pressure ratio, xV Figure 8: Plots of the BET adsorption isotherm per unit area using the BET area for nitrogen vapor adsorbing on α-alumina 13 and the zeta adsorption isotherm per unit area using As . The dark solid line represents the amount adsorbed per unit area calculated using the BET model and its reported area and the red line is for the adsorption amount per unit area determined using the ZAI model and As . The symbols indicate measured data per unit area, where the specific surface areas are determined using the respective BET and ZAI methods. The nSV values corresponding to the experimental data divided by the BET and ZAI areas, can be read using the left and right nSV axes respectively.

20


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The Near-Equilibrium Extension of the Thermodynamic Isotherm As seen in the previous section, the BET approach leads to the prediction of an infinite amount being adsorbed in the limit of xV approaching unity. Thus, it cannot be used to determine the isotherm expression for application in the Gibbs adsorption equation. By contrast, the results shown in Figures 5 to 7 indicate the proposed ZAI method leads to thermodynamic isotherms for N2 vapor adsorbing on α-alumina, carbon, and silica at 77K in their respective heterogeneity-free ranges. It is assumed that in the remainder of the xV range, the values calculated are the amount that would be adsorbed if the materials were heterogeneity-free in this xV range. By using the ZAI approach, the respective isotherms can be determined using eq (17) and the values of the isotherm constants listed in Table 1. With the availability of these isotherm equations, it becomes possible to combine nSV with the Gibbs adsorption equation 47,48 and construct the expression for the surface tension of the solid-vapor interface, γ SV (xV ), plus an integration constant. We propose a definition of wetting and use it to determine the integration constant. This will allow the value of γ S0 , the surface tension of a material in the absence of adsorption, to be predicted for each of the nine materials, and allow another examination of our supposition that the isotherm relation can be extended. The three isotherms of each material are of material samples with specific-areas that vary strongly in value as seen in Table 1. Nonetheless, if the extension is valid it should indicate that the value of γ S0 is the same for each material. We show this condition is satisfied to within 2% or less for each material.

The Wetting Transition When the system is in thermal equilibrium at 77 K, the adsorbate can be in equilibrium with the vapor for 0 ≤ xV ≤ 1. We assume the thermodynamic isotherm, eq (17) plus the isotherm constants listed in Table 1, are also valid in the near-equilibrium states. These

21



Page 22 of 40

states are defined as the steady, thermal disequilibrium states when the vapor is maintained saturated at T V , but the adsorbate and substrate are cooled to T S . The extent to which the adsorbate and substrate can be cooled relative to the vapor is assumed to be until the wetting transition occurs. (A schematic of an apparatus that produced this circumstance for heptane adsorbing on silicon was shown earlier. 24,27 ) The entropy of the adsorbate will be examined to identify the temperature of the substrate, TwS , required to reach the wetting transition. The wetting transition is defined after we define the dimensionless temperature-function, y V S . If T S is equal T V , then y V S = xV ,

(21)

but if the adsorbate and substrate are cooled below T V , then y V S is greater than unity and increases as T S is decreased further: yV S

Psat (T V ) ≡ Psat (T S )

TV

TS

when T S < T V .

(22)

If ywV S denotes the wetting condition, we suppose the states for 1 ≤ y V S ≤ ywV S may be approximated as near-equilibrium states. Then for these states, y V S may replace xV in the expressions for the cluster distributions a0 and aζ , and in the expression for nSV . 24 The calculated amount adsorbed as y V S is increased is shown in Figure 9, and is seen there to indicate that the amount adsorbed increases with y V S . As the amount of N2 vapor adsorbed is increased, other changes in the solid-vapor interface are induced. One of them is in the entropy of the adsorbate. Another is in the surface tension of the solid-vapor interface, γ SV . We examine each in turn. The canonical ensemble expression for the intensive entropy of the adsorbate, sSV , is

22


Page 23 of 40


300


250

α -alumina 15 nSV(µ mol/m2 )

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


200

150

100

α -1

50

0 0

0.2

0.4

x

V

0.6

0.8

1

1.2 VS 1.4 y

Figure 9: Amount of N2 vapor adsorbed per unit surface area on each of the three αalumina samples. 13–15 This corresponds to nSV as a function of xV in the equilibrium range and as a function of the temperature function, y V S in the disequilibrium range. Wetting transition occurs when y V S equals α−1 . The temperature of the substrate TwS to which the substrate must be cooled in order to initiate wetting, is listed in Table 2.

23



given in eq (11) which in the near equilibrium states becomes: 24,27 ! ζth X sSV (y V S ) aζ (y V S ) aζ (y V S ) =− ln . kB M M ζ=0

(23)

For each of the three samples of α-alumina, plots were made of sSV (y V S )/kB using eqs (6) and (7) and the isotherm constants listed in Table 1. These three plots are shown in Figure 10. Similar results were found for both carbon and silica. Each material is found to have a maximum entropy when y V S reaches their respective values of α−1 . Each sample of αalumina is found to have the same value of α−1 , suggesting α−1 is a property of α-alumina. As may be seen from Table 1, the same suggestion may be made for carbon and silica. The

N2 vapor adsorption on α -alumina

5

Thermal disequilibrium range

Equilibrium range

Non-dimensional entropy sSV/kB

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 24 of 40

Heterogeneity-free range

4

3

2

1 α -1

xV m

xV µ 0 0

0.2

0.4

xV

0.6

0.8

1

1.2 VS 1.4 y

Figure 10: Non-dimensional entropy of N2 vapor adsorbed on three α-alumina samples. 13–15 The adsorbate entropy increases to a maximum at y V S equals α−1 and then it decreases. This maximum point indicates that the value of α−1 , the wetting transition point of the adsorbate, is identical for all samples of α-alumina as seen in Table 1.

24


Page 25 of 40

maximum in sSV (y V S )/kB at α−1 indicates a possible phase transition in the adsorbate. 27 We investigate by considering cluster distributions that are shown in Figure 11. The fraction

N2 vapor adsorbing on α -alumina 0.35 0.3

ζ =1

0.25

aζ/M

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


ζ =2

0.2

0.15 0.1

ζ th =37

ζ =5 α -1

ζ =10

0.05

ζ =30

ζ =25 0 1

1.1

1.2

1.3

Temperature function, yVS

1.4

1.5

Figure 11: Cluster distribution of N2 vapor adsorbed on α−alumina samples. 13 The cluster distributions of all cluster types have the same value at y V S equals α−1 . The threshold number of molecules in a cluster at an adsorption site is ζth and it is finite.

of the adsorption sites per unit area, M , that are occupied by molecular clusters with ζ molecules is shown in this figure. At y V S equal unity, most of the sites are occupied by single molecules. A negligible number are occupied by clusters with the maximum number of molecules, ζth . As indicated in Figure 11, when y V S is equal α−1 , the number of adsorption sites that are occupied by one molecule is equal to the number of sites occupied by clusters with ζth molecules. If the amount adsorbed at α−1 is that shown in Figure 9, and it is converted into the average number of molecules adsorbed (by dividing nSV by M ) then at α−1 , on average, 25



Page 26 of 40

there would be 20 molecules adsorbed per site. We assume that at this condition, the larger clusters would contact smaller ones, as well as one another, and this contact would result in agglomeration of the clusters into an adsorbed liquid film. We note that agglomeration of molecular clusters was found to be the mechanism by which nanopores were filled when water and three hydrocarbons were each exposed to MCM-41 and to SBA-15. 49 Examination of the Wetting Definition At y V S equal α−1 , there is a liquid film of N2 adsorbed that is on average 20 molecules thick. We assume the surface tension of the substrate interface would have been lowered to the surface tension of the liquid-vapor interface, γ LV : γ SV (α−1 ) = γ LV .

(24)

The physical meaning of the wetting condition can be examined in two ways: firstly, if nitrogen vapor is saturated at T V and in contact with one of the nine powders, we first calculate the temperature of a substrate that would be necessary to bring about the wetting transition, denoted TwS . Secondly, we examine the wetting definition by using eq (24) with the near-equilibrium extension of the thermodynamic isotherm and the Gibbs adsorption equation 23,47 to calculate the surface tension of the solid-vapor interface, γ SV (y V S ). We use this expression for the surface tension to calculate γ S0 , and examine it to determine if the values obtained for the nine material samples indicate γ S0 is a material property. Wetting would be expected to be initiated when (α)−1 = y V S (T V , TwS ),

(25)

where T V is the temperature of the saturated N2 vapor that is in contact with a substrate, and TwS is the temperature to which the substrate must be cooled in order initiate the wetting transition: (α)−1

Psat (T V ) = Psat (TwS )

26

TV

S Tw

.


(26)

Page 27 of 40

The temperature of the vapor was taken to be that reported with the adsorption measurements. The thermodynamic properties of nitrogen 50 and the values of α listed in Table 1 were used with eq (26) to calculate TwS by iteration. The values found are listed in Table 2. Although the values of As for the nanopowders of each material vary by at least an order of magnitude (Table 2), the experimental values of α are approximately uniform for each material (Table 1) and the values of TwS predicted from eq (26) is found to be approximately uniform at 74.4 K. This indicates a substrate cooling of between 2.6 and 2.85 K is required to induce wetting of the nanopowders (Table 2).

N2 vapor adsorbing on α -alumina 70 60 50

MkB T ln( 1+c ζth )

γSV (mJ/m2 )

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


40 30 20 γ SV (α -1)=γ LV

10 0 0

0.2

0.4

0.6

0.8

1

xV

1.2

1.4

y

VS

Figure 12: Solid-vapor surface tension, γ SV , of nitrogen adsorbate on α−alumina samples. 13–15 When y V S equals α−1 , the surface tension of the solid-vapor interface equals the surface tension of the liquid-vapor interface, γ LV . In the absence of adsorption, when y V S equals zero, γ SV becomes the solid surface tension γ S0 .

27



Page 28 of 40

Calculation of the Surface Tension of the Solid-Vapor Interface as a Function of y V S The Gibbs adsorption equation of an isothermal system relates the thermodynamic isotherm to the surface tension of the solid-vapor interface dγ SV = −nSV dµSV ,

(27)

and if nitrogen vapor may be approximated as an ideal gas, the Gibbs adsorption equation becomes dγ SV = −

kB T nSV (y V S ) V S dy . yV S

(28)

The expression for nSV (y V S ) is obtained from the ZAI (eq (17)) by replacing xV by y V S . Then eq (28) may be integrated to obtain 1 + (c − 1)αy V S − c(αy V S )ζth +1 SV VS γ (y ) = C − M kB T ln , 1 − αy V S

(29)

where C is the integration constant. It may be evaluated by imposing eq (24) on eq (29), then C=γ

LV

1 + (c − 1)αy V S − c(αy V S )ζth +1 , + M kB T lim ln y V S →1/α 1 − αy V S

and after evaluating the limit C = γ LV + M kB T ln [1 + cζth ].

(30)

After combining eqs (29) and (30), the expression for γ SV (y V S ) in terms of the adsorption isotherm constants for a material sample is obtained as: (1 − αy V S )(1 + cζth ) SV VS LV γ (y ) = γ + M kB T ln . 1 + (c − 1)αy V S − c(αy V S )ζth +1

28


(31)

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By evaluating eq (31) at y V S equal zero, we obtain the expression for the surface tension of the solid substrate in the absence of adsorption: γ S0 = γ LV + M kB T ln(1 + cζth ).

(32)

The value of γ S0 for each of the nine material samples considered may be calculated using eq (32) and the values of c and ζth that are listed in Table 1. The values obtained for γ S0 are listed in Table 2. Plots of γ SV for the three samples of α-alumina are shown in Figure 12. For the three samples of carbon and for the three samples of silica, the calculated γ SV (y V S ) show similar results. We note that the expression of γ SV (y V S ) for each material sample was obtained from the integration of the Gibbs adsorption equation combined with the ZAI (eq (17)). The integration constant was evaluated by imposing the assumed wetting condition (eq (24)). Thus, based on these assumptions, the value of γ S0 obtained by evaluating γ SV (y V S ) at y V S equal zero is a prediction of a material property of the substrate, γ S0 . It is hence important that the values found for each material are consistent with one another. As seen in Table 2, the variation from the mean is 2% or less. Since σA was assumed to have the same value for N2 vapor adsorbing on any heterogeneityfree material sample, the value of M was predicted to be the same on any such material sample. See eq (18) and Table 1. Nonetheless, γ S0 is indicated to depend on the product of c and ζth . Ideally, this product would have had the same value for all samples of each material. Then the predicted values of γ S0 for each material would have been totally consistent, but the product cζth of the material samples did have slightly different values. Therefore, γ S0 of each material was found to vary by 2% or less.

DISCUSSION and CONCLUSIONS The expression for As has been developed and is given in eq (14). The expression may be applied if the cross-sectional area of an adsorbed N2 molecule, σA , and the number of 29



adsorption sites per unit mass, Mg , are known. We suppose σA has a value of 0.162 nm2 when N2 vapor at 77 K adsorbs on each of the nine material samples considered. And to determine Mg , we use the mass-specific ZAI (mZAI) eq (12), as the model function in a nonlinear regression analysis with the measured adsorption data in the xV -range where each material sample is indicated to be heterogeneity-free, see Figures 4 and 5. From this procedure, we determine the values of the constants appearing in the mZAI expression (Mg , c, α, and ζth ) that minimize the error for each material sample. We then show the thermodynamic isotherms can be used to calculate the entropy of the adsorbate, the cooling of the substrate required to bring about wetting, and the surface tension of the solid-vapor interface from wetting, γ SV (α−1 ), to the absence of adsorption, γ S0 . We first formulate the ZAI expression for the thermodynamic isotherm for each material sample. Application of eq (14) then gives the values of As for each material sample. The values are listed in Table 1. Note that the three values of As for the samples of each material (α-alumina, carbon, and silica) vary over a wide range. But, independently of the value of As , when M c, α, and ζth are used with the ZAI (eq (17)) to calculate the three thermodynamic isotherms for each material sample, the calculated isotherms very nearly coincide, and the data for each material similarly agree with the calculated isotherm, See Figures 5 to 7. Although it is oft assumed that σA has a value of 0.162 nm2 for all materials, 15,51,52 it is still controversial. 40,53,54 Our results support the notion that σA has one value for nine samples of the three materials considered. But we are unable to claim support for any particular value of σA . The same consistency in the thermodynamic isotherms would have been seen for any reasonable value of σA . In the proposed approach, the heterogeneity-free range of xV is identified using eq (12). This range is defined by two constants: xVµ and xVm . In the very low pressure range (0 ≤ xV ≤ xVµ ), some of the measured adsorption data do not agree with the calculations made with mZAI eq (12). This disagreement is assumed to result from heterogeneities, such as micropore filling. 29,35 Similarly, some high pressure data points (those for xVm ≤ xV ≤ 1) also

30


Page 30 of 40

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deviates from the mZAI, possibly due to capillary condensation. 16,36 An iterative procedure was developed to identify the values of xVµ and xVm from the reported data of N2 adsorbing on α-alumina, 13–15 carbon, 16–18 and silica. 19–21 For example, the different values of xVµ and xVm for three samples of α-alumina are indicated for Figure 4. Apparently, the values of these heterogeneity limits depend on the slightly different manufacturing techniques used to produce the different material samples. As seen in Figures 5 to 7, the thermodynamic isotherm for each material, i.e., the ZAI, is consistent with the data in the heterogeneity-free range of xV . And since the ZAI, eq (17), indicates a finite amount to be adsorbed when xV is equal to unity, 5,26 it can be extended outside the heterogeneity range . The validity of these predictions were investigated by introducing the temperature function y V S , eq (22). It was used to consider thermal-disequilibrium states in which the solid substrate was reduced to T S while the N2 vapor was maintained saturated at T V . 24,27 These states were approximated as near-equilibrium states, and the ZAI was applied to obtain an expression for the entropy of the adsorbate, sLV as a function of y V S . The entropy function sLV (y V S ) indicates a phase transition in the adsorbate when y V S reaches α−1 . Since α had first appeared as a mZAI-constant, its value had already been determined and was listed for each material sample in Table 1. Thus, eq (26) could be applied to calculate TwS . The subcooling required to initiate wetting, T V − TwS , was found to be 2.6 K for each α-alumina sample, approximately 2.5 K for the carbon samples, and approximately 2.8 K for the silica samples. When thermal equilibrium exists, the estimated maximum number of molecules per site is 2 to 3 for α-alumina, but when the substrate is cooled to wetting, a liquid film is estimated to form that is 20 molecules thick. The surface tension of the solid-vapor interface is assumed to be transformed to that of the liquid-vapor interface. This assumption provided an essential boundary condition for the prediction of the surface tension of each material sample in the absence of adsorption, γ S0 , a material property.

31



After assuming the ZAI, eq (17), with the isotherm constants listed in Table 1 described the amount adsorbed in the entire (including the heterogeneity) range : 0 ≤ y V S ≤ α−1 , the nSV was used to integrate the Gibbs adsorption equation. This led to an expression for γ SV that included an integration constant. It was evaluated by requiring γ SV to satisfy the wetting condition, eq (24). As a result, the complete expression was obtained for γ SV (y V S ) for each material sample. When this expression was evaluated at y V S equal zero, the values of γ S0 listed in Table 2 were found. Note the consistency of the predicted γ S0 values for each material. They differ by 2% or less.

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

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use for characterization of mesoporous alumina-based materials. Adsorption 2013, 19, 475–481. ˇ (15) Matˇejov´a, L.; Solcov´ a, O.; Schneider, P. Standard (master) isotherms of alumina, magnesia, titania and controlled-pore glass. Microporous and Mesoporous Materials 2008, 107, 227–232. (16) Kruk, M.; Jaroniec, M.; Gadkaree, K. P. Nitrogen Adsorption Studies of Novel Synthetic Active Carbons. Journal of Colloid and Interface Science 1997, 192, 250 – 256. (17) Silvestre-Albero, A.; Silvestre-Albero, J.; Mart´ınez-Escandell, M.; Futamura, R.; Itoh, T.; Kaneko, K.; Rodr´ıguez-Reinoso, F. Non-porous reference carbon for N2 (77.4K) and Ar (87.3K) adsorption. Carbon 2014, 66, 699 – 704. (18) Kruk, M.; Li, Z.; Jaroniec, M.; Betz, W. R. Nitrogen Adsorption Study of Surface Properties of Graphitized Carbon Blacks. Langmuir 1999, 15, 1435–1441. (19) Jaroniec, M.; Kruk, M.; Olivier, J. P. Standard Nitrogen Adsorption Data for Characterization of Nanoporous Silicas. Langmuir 1999, 15, 5410–5413. (20) Bhambhani, M.; Cutting, P.; Sing, K.; Turk, D. Analysis of nitrogen adsorption isotherms on porous and nonporous silicas by the BET and αs methods. Journal of Colloid and Interface Science 1972, 38, 109 – 117. (21) Baker, F. S.; Sing, K. S. W. Specificity in the Adsorption of Nitrogen and Water on Hydroxylated and Dehydroxylated Silicas. J. Colloid Interface Sci. 1975, 55, 605. (22) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover, New York, 1961; Vol. 1; pp 55–349. (23) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans, Green and Co.: London, 1996; p 24.

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1

TOC IMAGE

Figure 13: TOC Image

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N2 vapor adsorbing on , -alumina 0.4

50

, -alumina 13

nSV(7 mol/m2 )

, -alumina 14

nSV (7 mol/mg) 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

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, -alumina 15

0.2

0

Heterogeneity-free range 25

xV7

xVm

0 0

0.5

1

0



0.5

1

Specific Surface Area, Wetting, and Surface Tension of Materials From

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