Thin Solid Films 519 (2011) 2994–2997

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Validity of Lorentz–Lorenz equation in porosimetry studies

Daniel Schwarz ⁎, Herbert Wormeester, Bene PoelsemaIMPACT Institute, Universiteit Twente, Postbus 217, 7500 AE Enschede, The Netherlands

⁎ Corresponding author. Tel.: +31 534893109; fax: +E-mail address: d.schwarz@tnw.utwente.nl (D. Schw

0040-6090/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.tsf.2010.12.053

a b s t r a c t

a r t i c l e i n f o

Available online 16 December 2010

Keywords:Ellipsometric porosimetryEffective medium approximation

Ellipsometric porosimetry is a valuable tool to determine gas loading of porousmaterials. Usually the Lorentz–Lorenz effective medium theory is used, instead of the more accurate Bruggeman theory. In contrast toLorentz–Lorenz, the Bruggeman model requires detailed knowledge on the constituents of the porousmaterial. A first order perturbation of both effective medium approximations is used to analyze the differencebetween these models. Similar results are only found for materials with 70% porosity. Below 50% porosity, thegas load is underestimated with the Lorentz–Lorenz model. For porous silica and alumina with 50% porosity,the use of Lorentz–Lorenz leads to a systematic error of 18% of the load capacity.

31 534891101.arz).

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

In ellipsometric porosimetry the loading of a porous host materialwith a guest material (a gas or liquid) is studied, for example the CO2

sorption of a silica membrane [1]. In this technique the change of thedielectric function upon loading is measured. From this change, theamount of guestmolecules in thehostmaterial or thematerial's porositycan be calculated. To do these calculations, the effect of the relativepresence of the host and guest material dielectric function has to beevaluated from an effective medium approximation (EMA).

Usually a Lorentz–Lorenz (also referred to as Clausius–Mossotti)approach is used, instead of a generally more accurate Bruggemanapproach [2–4]. The reason for this lies in the often unknown dielectricproperties of the constituents of the porous material. For example,porous silica can often not be represented as a mixture of silicon oxideand voids due to the presence of many hydrogen bonds. The SiOHmaterial leads to a higher dielectric function than quartz [1].

If the guest material has a small dielectric constant, which is usuallythe case for a dilute gas, the loading of the host material will result in asmall change in the total dielectric function. This means that it'sexpected to be possible to describe the change by a first orderperturbation. This linearization is done for both effective mediumtheories in this article and the result is compared. A significant deviationbetween the two is found for low porous materials.

2. Effective medium approaches

2.1. Lorentz–Lorenz

The Lorentz–Lorenz equation is derived from the Clausius–Mossotti relation, which relates the dielectric constant of spherical

particles with their densityN and their polarizability α [8,9]. In SI unitsit is:

⟨ε⟩−1⟨ε⟩ + 2

=Nα3ε0

ð1Þ

For a mixture of several materials with polarizability αi and densityNi, the contributions of the individual components are counted up togive the effective dielectric function ⟨ε⟩ of the mixture. This approachwas originally derived by Lorentz and Lorenz to describe the opticalproperties of a gas, a case in which the single molecules are wellseparated, and do not interact.

⟨ε⟩−1⟨ε⟩ + 2

=13ε0

∑iNiαi ð2Þ

A linearization for a dilute gas (εg≈1) simplifies the expression to[8]:

εg = 1 +Ngαg

ε0ð3Þ

For a porous material with dielectric function εm and porosity f, theLorentz–Lorenz is often used to describe the change in dielectricconstant through the insertion of a gas with polarizability αg into thepores. The gas is assumed to fill all pore volume homogeneously, and itsdensityNg is increased upon loading. Eq. (2) can be adapted for the twocomponents, the solid material, and the added gas to give the Lorentz–Lorenz equation [5–7]:

⟨ε⟩−1⟨ε⟩ + 2

= 1−fð Þ εm−1εm + 2

+ fεg−1εg + 2

ð4Þ

2995D. Schwarz et al. / Thin Solid Films 519 (2011) 2994–2997

Since the density in this system can be much higher than for a gas,the assumption of well separated and non-interacting molecules isnot fulfilled in this approach, stretching the validity of the equation.

Eq. (4) can be rearranged to give the total dielectric function ⟨ε⟩:

⟨ε⟩ =2f εg + εgεm−2f εm + 2εm

εg + f εm−εg� �

+ 2ð5Þ

2.2. Bruggeman effective medium approximation

Bruggeman calculated in his famous article [2] the dielectricconstants for mixed media of different dimensionality and topology.Usually his result for media made up of spheres of two materials iscalled the Bruggeman approach and is probably the most commonEMA used in ellipsometry. By integrating the Rayleigh mixing formulafor two components Bruggeman calculated the expression for thedielectric constant ⟨ε⟩.

Adapted for the case of a host material with spherical pores whichare filled with guest molecules of density Ng, the effective dielectricconstant ⟨ε⟩ is given by:

0 = 1−fð Þ εm−⟨ε⟩εm + 2⟨ε⟩

+ fεg−⟨ε⟩

εg + 2⟨ε⟩ð6Þ

Fig. 1 shows the solutions for a material with dielectric constantεm=2.1 and different porosities as a function of guest permittivity εgfor the Lorentz-Lorenz and Bruggeman equation. Especially for guestswith a small εg, i.e. a very dilute gas, and a porosity of 10% to 60% bothapproaches differ. This signifies that the choice between Lorentz–Lorenz and Bruggeman in ellipsometric porosimetry is in this regionsignificant for the obtained result. Because the Bruggeman approach isregarded to be the more accurate one, the use of the Lorentz–Lorenzapproach leads to a systematic error.

3. Linearization of the EMA

If the dielectric constant of the guest material is small, the loadingis expected to result in a small change of the dielectric constant. Thisguest material could be for example a dilute gas. To estimate thechange of the dielectric constant it is therefore sufficient to describe itby a first order perturbation. According to Eq. (3) the change isexpected to depend on the polarizabilityαg and the guest densityNg. Ifthe dielectric constant of the empty host material is ⟨ε⟩0, the change

2.00

1.75

1.50<ε>

1.25

1.00

1.0 1.2

Guest dielectric constant εg 1.4 1.6

Bruggeman

f = 0

f = 0.5

f = 1

Lorentz-Lorenz

1.8 2.0

Fig. 1. Solution for the effective dielectric constant ε as a function of εg and for porosity ffrom 0 to 1 (line separation 0.1) for the Lorentz-Lorenz (dash) and Bruggeman (solid)equation. The host material has εm=2.1.

upon gas loading is expressed by a term δε:

⟨ε⟩ = ⟨ε⟩0 + δε = ⟨ε⟩0 + CfNgαg

ε0ð7Þ

Where the product fNg is the guest molecule density inside thepores. The linearization coefficient C depends on the effectivemediumapproach employed. For the Bruggeman and Lorentz–Lorenz ap-proach the coefficients CB and CL are respectively (see Appendix forderivation):

CB =1f

3f−1ð Þ⟨ε⟩0 + εm2⟨ε⟩0 + εm

⟨ε⟩

ð8Þ

CL =⟨ε⟩0 + 2ð Þ2

9ð9Þ

Note that for the Bruggeman approach both the porosity f and thehost material dielectric constant εm are needed, quantities which areusually not known. On the other hand for the Lorentz–Lorenzapproach, only the effective dielectric function of the empty, porousmaterial is needed, which can be easily measured.

3.1. Validity of the linearizations

To test the validity of the linearizations, they were compared withthe exact results for the parameters of a porous silica material (εm=2,f=0.5) loaded with a dilute gas (εg=1.001), a dense gas (εg=1.1)and a liquid (εg=1.8).

The effective dielectric function is calculated as a function of therelative concentration Ng/Nfull, which corresponds to an increasingguest density inside the pores.

Fig. 2 shows the comparison between the exact solution for theLorentz-Lorenz approach and its linearization calculated with Eq. (9).While the linearization provides a good representation for theinsertion of a dilute gas, the insertion of a denser gas or even of aliquid results in a substantial deviation between the linearized and theexact evaluation.

In Fig. 3 the exact and linearized results for the Bruggemanapproach are presented, using the same parameters as in Fig. 2. Theresults show that the linearization gives a very good representation ofthe exact solution. Even for a very dense guest material (εg=1.8) thelinearization is still acceptable up to 50% relative concentration.Therefore, the loading of a porous material can be well described bythe linearized version of Bruggeman's equation.

<ε>

2.2

2.0

1.8

1.6

1.4

0.0 0.2

relative guest concentration

εg =1.8

εg =1.1

εg =1.001

0.4 0.6

Lorentz-Lorenz

Linearization of Lorentz-Lorenz

0.8 1.0

Fig. 2. Comparison of the exact and linearized solution for the Lorentz-Lorenz approach.Three different guest materials with εg=1.8 (liquid), 1.1 and 1.001 (gas) are used.

<ε>

2.0

1.8

1.6

1.40.0 0.2

relative guest concentration

εg =1.8

εg =1.1

εg =1.001

0.4 0.6

Bruggeman

Linearization of Bruggeman

0.8 1.0

Fig. 3. Comparison of the exact and linearized solution for the Bruggeman approach.Three different guest materials with εg=1.8 (liquid), 1.1 and 1.001 (gas) are used.

<ε> 0 = 1

<ε> 0 = 3

porosity f

100

x (C

B -

CL)

/ C

B

0.0

40

20

0

-20

-400.2 0.4 0.6 0.8

Fig. 5. Normalized difference between CB and CL as a function of porosity. Calculated forvalues of the empty material dielectric constant ⟨ε⟩0=1 to ⟨ε⟩0=3 (line separation 0.2,εg=1.001).

2996 D. Schwarz et al. / Thin Solid Films 519 (2011) 2994–2997

4. Discussion

4.1. Influence of the host material

To compare the influence of the choice between the Lorentz–Lorenz and Bruggeman approach for porosimetry measurements, thelinearization coefficient C is calculated while changing diverseparameters.

In Fig. 4 the linearization coefficient C is shown as a function of thedielectric constant bεN0 of the empty, porous host for porosities fromf=0.1 to f=0.8. A strong influence of the actual value bεN0 on theresulting coefficient for both approaches is observed. The Lorentz–Lorenz approach has the advantage that this is the only parameterthat determines the linearization coefficient, however it is totallyindependent of the actual material's porosity. The Bruggemanlinearization shows that the value of the linearization coefficient isactually considerably smaller for low porosity materials.

For example for the parameters of porous silica (bεN0=2, f=0.5)loaded with a CO2 gas, the use of the Lorentz–Lorenz equation wouldlead to a 18% underestimation of the amount of CO2 adsorbed. Fig. 5presents the normalized difference between both coefficients as afunction of porosity f. Only for very porous materials (f≈0.7) theLorentz–Lorenz and Bruggeman approach give approximately thesame result. A strong systematic deviation occurs for less porousmaterials.

<ε>0

5

4

3

2

1

1.0 1.5 2.0 2.5

f = 0.1

f = 0.8

3.0

Coe

ffici

ent C

Bruggeman

Lorentz-Lorenz

Fig. 4. Linearization coefficient C for the Lorentz-Lorenz and Bruggeman approach as afunction of empty material dielectric function ⟨ε⟩0 (εg=1.001).

5. Conclusion

The Lorentz–Lorenz and Bruggeman equations were linearized todescribe the insertion of a dilute guest material into a porous hostmaterial. A comparison of the resulting linearization coefficientsshows that the Lorentz–Lorenz equation can only be safely quanti-tatively applied for very porous materials (f≈0.7) in porosimetrystudies. Usually it will result in a substantial underestimation ofaround 20% of the amount of gas adsorbed.

Appendix A. Calculation of linearization coefficient

The coefficient C is calculated by adding a small term δεbb1 to thetotal dielectric constant, and solving the respective equation for thepolarizability αg of the guest molecule. The derivation is given forcompleteness.

Appendix A.1. Lorentz–Lorenz

Modifying Eq. (4) by adding δε gives:

⟨ε⟩0 + δεð Þ−1⟨ε⟩0 + δεð Þ + 2

= 1−fð Þ εm−1εm + 2

+ fεg−1εg + 2

ðA:1Þ

Subtracting from both sides the situation of the empty hostmaterial:

⟨ε⟩0 + δεð Þ−1⟨ε⟩0 + δεð Þ + 2

− ⟨ε⟩0−1⟨ε⟩0 + 2

= fεg−1εg + 2

ðA:2Þ

The right side can be simplified by applying Eq. (3) for εg≈1.Expanding the left side and solving for δε gives then:

δε =⟨ε⟩0 + 2ð Þ2

9fNgαg

ε0ðA:3Þ

Comparing this result with Eq. (7) gives the coefficient CL for theLorentz–Lorenz equation.

CL =⟨ε⟩0 + 2ð Þ2

9ðA:4Þ

2997D. Schwarz et al. / Thin Solid Films 519 (2011) 2994–2997

Appendix A.2. Bruggeman

[3] Eq. (6) needs to be modified to include δε:

f−1ð Þ εm− ⟨ε⟩0 + δεð Þεm + 2 ⟨ε⟩0 + δεð Þ = f

εg− ⟨ε⟩0 + δεð Þεg + 2 ⟨ε⟩0 + δεð Þ ðA:5Þ

Solving this equation for δε gives:

δε =εgεm + ⟨ε⟩0 3f−1ð Þεg + εm 2−3fð Þ−2⟨ε⟩0

� �

εm 3f−2ð Þ−εg 3f−1ð Þ + 4⟨ε⟩0ðA:6Þ

The nominator can be reduced by using the fact that for εg=1,δε=0. This leads to the simplification:

δε =εg−1

� �ðεm + 3f−1ð Þ⟨ε⟩0

εm⟨ε⟩0

+ 2⟨ε⟩0 + 3f−1ð Þ 1−εg� � ðA:7Þ

Comparing this result with Eq. (7) and omitting the εg−1 term inthe denominator gives the coefficient CB for the Bruggemanequation. [4]

CB =1f

3f−1ð Þ⟨ε⟩0 + εm2⟨ε⟩0 + εm

⟨ε⟩0

ðA:8Þ

References

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