Dynamic pressurization: novel method for measuring fluid permeability

Journal of Non-Crystalline Solids 325 (2003) 34–47 www.elsevier.com/locate/jnoncrysol

Dynamic pressurization: novel method for measuring fluid permeability Joachim Gross, George W. Scherer

*

Princeton Materials Institute, Princeton University, Eng. Quad E-319, Princeton, NJ 08544, USA Received 25 February 2003

Abstract The permeability of an aerogel can be determined during the process of supercritical drying by measuring the dilatation of the body as the pressure in the autoclave is changed. When the pressure is reduced in the autoclave by venting the supercritical fluid, the higher pressure inside the aerogel causes it to expand; the strain relaxes as the fluid flows out of the gel. We present an analysis of the the kinetics of this volumetric strain and apply it to a silica aerogel. The linear strain of the gel was measured with an LVDT mounted inside the pressure vessel. The permeability was obtained over a range of pressure from about 8 MPa to ambient pressure, and was found to vary inversely with the pressure, as expected from the Klinkenberg model. The permeability obtained by extrapolation to infinite pressure agreed with the independently measured liquid permeability obtained by beam-bending. Ó 2003 Elsevier B.V. All rights reserved.

1. Introduction Aerogels are materials with extremely high porosity (typically >90%, often >99%), but with mean pore sizes in the mesoporous range (i.e., radii 1 6 r ðnmÞ 6 25). Usually, aerogels are made by supercritical drying (SCD) of gels [1], but structurally equivalent materials have been produced by other methods [2]. Measurement of the permeability, D, of aerogels to fluid is important for practical and theoretical reasons. First, rapid changes in temperature [3] or pressure [4] during SCD can cause cracking of the sample; to avoid such damage, it is important to develop a model of

*

Corresponding author. Fax: +1-609 258 1563. E-mail address: [email protected] (G.W. Scherer).

the process, which requires knowledge of D as a function of the pressure, PA , in the autoclave. Second, the interaction between the solid network and the pore fluid is central to certain applications of aerogels, such as gas filters and acoustic couplers [5,6]. Finally, the small pores of aerogels lead to a transition from laminar flow to Knudsen diffusion as the pressure decreases, so these materials are useful for testing theories of transport in porous materials. The Wurzburg group measured gas transport in aerogels by imposing stepwise changes in pressure, then measuring the kinetics of relaxation of the pressure in the chamber as gas diffused into the sample [5,6]. The method is limited to low pressures, because large pressure jumps can cause distortion of the apparatus that compromises the accuracy of the measurement. In fact, that problem

0022-3093/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-3093(03)00359-4

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

was later found to have affected the data presented in [5]; it was avoided in later studies [6]. By combining transient permeability measurements obtained by the pressure jump technique with static flow measurements, they were able to estimate the tortuosity of flow path and the degree of gas adsorption on the surface [8]. Recently, Gross [9] introduced a different method that is applicable to a wider range of pressure. He showed that the data could be obtained during the SCD process itself, by measuring the deformation of the sample resulting from stepwise changes in the autoclave pressure, PA . Since the method involves a direct measurement of the strain of the sample, it is not influenced by any dilatation of the pressure chamber. He presented a detailed analysis of the deformation, taking full account of the non-ideal behavior of the fluid (in this case, CO2 ), and was able to calculate the permeability of a silica aerogel as a function of pressure from ambient to about 10 MPa. In that paper, Gross treated the pressure jumps as adiabatic, but later concluded (as explained in Appendix A) that the experimental conditions were actually isothermal. In the present paper, we present a simplified analysis of the deformation produced by an isothermal jump in pressure and use it to obtain the permeability as a function of pressure. Our values of D are close to those obtained by Gross [9], but the calculation is simpler. Of course, the experimental method, called dynamic pressurization (DP), is not limited to aerogels. Any material that is sufficiently compliant to be deformed by small pressure differentials could be examined using DP. With the analysis presented below, one can estimate the maximum deformation to be expected, given the size of the pressure jump and the elastic modulus of the body, then determine whether that strain would be measurable with the equipment available.

2. Experimental procedure Measurements were performed on a silica gel made from tetraethoxysilane with the B2 recipe [10], a two step process using acid hydrolysis and base-catalyzed condensation. After mixing, the sol

35

was cast in a polystyrene pipette tube (7.8 mm diameter, 10 cm long); it gelled after about 5 h, was aged for four days, then washed 7 times over 9 days in tetrahydrofuran (THF). One gram of trimethyl chlorosilane (TMCS) was added to the test tube and allowed to react with the gel for 24 h; this makes the final gel hydrophobic. Finally, the gel was washed twice more with THF and then exchanged repeatedly with pure ethanol. This recipe yields a gel with a relative density (or, volume fraction of solids) q of 0.068. A beam-bending experiment was performed in ethanol on a gel rod with a span/diameter ratio > 10. When a fixed deflection is applied to the rod of gel, a pressure gradient is created in the liquid within the pores. As liquid flows to equilibrate the pressure within the gel, the force that must be applied to sustain a constant deflection decreases with time. Fitting the measured force to the theoretical curve [11–13] yields YoungÕs modulus, Ep , PoissonÕs ratio, mp , and the characteristic relaxation time, s s¼

2ð1
mp ÞgF R2 ; D1 Hp

ð1Þ

where gF is the viscosity of the pore fluid, R is the radius of the rod, D1 is the liquid permeability, and Hp is the longitudinal modulus Hp ¼

ð1
mp ÞEp : ð1 þ mp Þð1
2mp Þ

ð2Þ

SCD was performed by placing the gel in an autoclave and flushing with liquid CO2 , then raising the temperature and pressure above the critical point, and slowly depressurizing. If there were no shrinkage, the aerogel density would be 135 kg/m3 , which corresponds to q ¼ 0:068 and a porosity of 93%. However, syneresis and shrinkage during SCD increased the density to 200 kg/m3 , or q ¼ 0:10, as determined by measuring the aerogel dimensions and weighing. The longitudinal sound velocity in the aerogel was measured using two 180 kHz piezotransducers. The acoustic wavelength was about 1 mm, so this measurement yields the longitudinal modulus, Hp . The BET surface area of an aerogel was measured by nitrogen adsorption (Micromeritics 2010).

36

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47 LVDT coil CO2 input valve

LVDT core

sample

from high pressure pump

exhaust valve to exhaust

T,p autoclave

Fig. 1. Computer-controlled autoclave fitted with an LVDT to measure dimensional changes of the gel sample.

For the dynamic pressurization experiments, a piece of the aerogel 66 mm in length and 6.9 mm in diameter was mounted in a sample holder made from stainless steel wires and suspended from the lid of the autoclave, as shown in Fig. 1. On top of the sample, a small piece of thin aluminum sheet was placed to protect it from being damaged. The core of a linear variable differential transformer (LVDT) displacement sensor was supported by a thin piece of stainless steel wire standing on the aluminum sheet. The LVDT coil was mounted on the outside of a piece of 6.35 mm high-pressure tubing connected to the autoclave lid and sealed at the upper end. This allows the axial extension or compression of the gel to be measured without the need for electrical connections to the interior of the pressure vessel. The wire and core together weighed less than 3 g, exerting a stress of about 1 kPa on the gel cross-section. This caused a bias axial strain of )0.1%, small enough to be neglected. The autoclave was heated with a water pipe coil connected to a recirculator running at 55 °C. After sealing the autoclave, the input control valve was pulsed open for less than a second at

a time and pressure and strain data were logged continuously at a rate of 9 Hz. The sample deformation typically relaxed within about 10 s; after about 20 s the next step was initiated. Between pressure steps, the temperature in the vessel was also recorded by means of a thermocouple. This was repeated until the pressure in the vessel had reached the maximum pressure of 10.2 MPa, then the exhaust valve was used to control pressure steps of opposite direction until the autoclave returned to ambient pressure. Only the pressurization data are used in the following analysis, because the sample cracked during depressurization. In total, about 240 steps were recorded; each step was fitted separately. The compressibility and viscosity of CO2 were calculated from the equations recommended by Vukalovich and Altunin [14] at the temperature and pressure measured for each step.

3. Analysis of deformation In this section we analyze the deformation of a cylindrical porous body as a result of changes in the pressure in the surrounding vessel. The approach, which closely follows that of [4], is based on BiotÕs constitutive model for saturated porous elastic materials [15]. The constitutive equation for an isotropic, saturated porous body can be written in the following form:
z ¼
f þ

1 ½rz
mp ðrr þ rh Þ; Ep

ð3Þ

where
f ¼

PF þ
s : 3Kp

ð4Þ

In cylindrical coordinates, the total stresses (which represent the sum of the forces on the solid and liquid phases) are rr , rh , and rz ; the corresponding strains are er , eh , ez . The pressure in the fluid is PF ; Kp ¼ Ep =½3ð1
2mp Þ is the bulk modulus and es is the spontaneous baseline strain. We found that there was a reproducible strain of the gel, shown in Fig. 2, that depended only on the pressure. The origin of the strain is not certain, but the dried gel

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

had a noticeable odor of ethanol, so we suspect that the surface energy of the network was influenced by an adsorbed ethanol/CO2 solution. Compliant materials with high surface area typically show significant strains during adsorption [16]. The experiments indicate that
s is an increasing function of the pressure in the fluid (presumably owing to a corresponding change in composition of the adsorbed layer). This strain is treated as a linear function in the analysis, because the variation in pressure during any given pressure jump is small. The volumetric strain of the gel is given by Eq. (28) of [4]
¼ 3b
f þ 3ð1
bÞh
f i


PA ; Kp

bP_ F þ ð1
bÞhP_ F i
P_ A ; Kp

ð7Þ

where the superscript dot represents the partial derivative with respect to time.

2

1. 5

εs ( % )




q_ F þ


  qF 1 qF D r rPF ;
_ ¼ 1
q gF 1
q

ð8Þ

where gF is the viscosity and qF is the density of the fluid; the flux, J , of fluid within the body is assumed to obey DarcyÕs law [17] J ¼


D rPF ; gF

ð9Þ

where D is the permeability (units of area). If it is permissible to ignore the spatial variation of qF within the body, then Eq. (8) reduces to Eq. (12) of [4] ð1
qÞ

where u ¼ r=R and R is the radius of the cylinder. The volumetric strain rate is
_ ¼

The continuity equation is given by Eq. (10) of [4]

ð5Þ

where PA is the pressure in the autoclave and b ¼ ð1 þ mp Þ=½3ð1
mp Þ. The angle brackets represent the volumetric average for any function f ðr; hÞ RR Z 1 f ðr; hÞ2pr dr 0 ¼2 f ðu; hÞu du; ð6Þ hf ðhÞi ¼ pR2 0

37

D q_ F þ
_ ¼ r2 PF : gF qF

ð10Þ

This approximation will be justified shortly. We define the function
 qF ðhÞ XðhÞ ¼ ð1
qÞKp ln ; ð11Þ qF ð0Þ which is assumed to be a function only of time (i.e., the spatial variation of density within the sample is neglected). The spontaneous strain,
s , is approximated in a narrow pressure interval by
s ¼
0 þ ahPf i;

ð12Þ

where
0 and a are constants. With these approximations, Eq. (10) becomes
 1 o oPF oPF 1 dX u ð1 þ 3Kp aÞ þ ¼ u ou b dh ou oh
 1
b dhPf i þ ð1 þ 3Kp aÞ b dh 1 dPA
; ð13Þ b dh where the reduced time is defined by h ¼ t=sD , and the hydrodynamic relaxation time is defined as

1

sD ¼

0. 5

0 0

2

4

6

8

10

PF ( MPa )

Fig. 2. Reversible strain of gel as function of CO2 pressure.

bgF R2 : DKp

ð14Þ

The cylinder is assumed to be long compared to its radius, so that axial flow can be neglected; therefore Eq. (13) does not include axial gradients. For the sample used in the experimental study, Kp a  0:02, so it is a good approximation to

38

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

neglect the terms involving a. This reduces Eq. (13) to
 1 o oPF oPF df ðhÞ u ; ð15Þ þ ¼ u ou dh ou oh where we define df ðhÞ 1 dX ¼ þ dh b dh




1
b b



dhPf i 1 dPA
: dh b dh

ð16Þ

The initial condition in the experiments does not correspond to equilibrium, because the pressure is drifting upward or downward before and after each jump. For the moment, though, let us assume that the initial pressure in the sample, PF ðu; 0Þ, is equal to that in the autoclave, PF ðu; 0Þ ¼ PA ð0Þ. The boundary condition is PF ð1; hÞ ¼ PA ðhÞ. Subject to these conditions, Eq. (15) is solved by applying the Laplace transform with respect to h, which is defined by [18] Z 1 f~ðu; sÞ ¼ f ðu; hÞe
sh dh; ð17Þ 0

where s is called the transform parameter. The solution of Eq. (15) is Z h dX hPf ðhÞi ¼ PA ðhÞ
/ðh
h0 Þ 0 dh0 : ð18Þ dh 0

pffiffiffi Mathematicaâ [19]. Although h blows up as 1= h when h approaches zero, / is a well-behaved function that is closely approximated by 
  4b /ðhÞ  exp
pffiffiffi ðh1=2 þ 0:9603h þ 1:319h3=2 Þ : p ð22Þ This approximation providespthe ffiffiffi correct behavior as h ! 0 : / ! 1
ð4b= pÞh1=2 . The exact function and this approximation are shown in Fig. 3. Returning now to the question of the initial condition in the experiments, let us see what the steady-state pressure is inside the sample when the pressure changes continuously. If dX=dh ¼ A ¼ constant, then as h ! 1, Eq. (18) leads to lim ðhPf ðhÞi
PA ðhÞÞ ¼ lim
s/~ðA=sÞ:

h!1

Since lims!0 h~  1
s=8
lim ðhPf ðhÞi
PA ðhÞÞ ¼


h!1

1
h~ ; s½1
ð1
bÞh~

 A : 8b

ð24Þ

Therefore, if the pressure at the start of the experiment (i.e., just before the jump) has been

The Laplace transform of the relaxation function / with respect to h is /~ ¼

ð23Þ

s!0

1

ð19Þ

0.8

where 1 X

0.6

4 expð
B2n hÞ:

ð20Þ

n¼1

Here, Bn is a root of the Bessel function of the first kind of order zero, J0 ðBn Þ ¼ 0. The transform of h is pffiffi 1 X 4 2I1 ð sÞ ~ ð21Þ ¼ pffiffi pffiffi : h¼ s þ B2n s I 0 ð sÞ n¼1 The second equality in Eq. (21) is explained in Appendix B. The transform in Eq. (19) must be inverted numerically; this was done using the Stehfest algorithm in the Numerical Inversion package in

φ

hðhÞ ¼

0.4

Exact Approx

0.2

0 10-4

10-3

10-2

10-1

100

101

θ

Fig. 3. The dashed curve represents the exact function, /, whose Laplace transform is defined in Eq. (19), and solid curve is the approximation given in Eq. (22).

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

dent measurement of the elastic modulus, the permeability is found from Eq. (14). If an instantaneous step change in pressure of DPF is made, then Eq. (28) indicates that the instantaneous strain will be

100

10-1

ez  Ψ(θ)

39

DPF : 3Kp

ð29Þ

The preceding analysis can be used for any material that is sufficiently compliant to allow strains of this magnitude to be measurable for practical values of DPF .

10-2

10-3

4. Properties 10-4 10-4

10-3

10-2

θ

10-1

100

101

Fig. 4. The dashed curve represents the exact function, W, defined in Eq. (C.7), and the solid curve is the approximation given in Eq. (C.8).

changing steadily at a rate corresponding to A, the initial condition is
 A hPf ð0Þi ¼ PA ð0Þ
: ð25Þ 8b

The data were fitted to Eqs. (12), (22), (26), and (28) using a simplex routine [20], with sD as the only free parameter. The property data for CO2 were taken from equations given by Vukalovich and Altunin [14]. The temperature varied from about 33 °C to 45 °C during the experiment as the pressure rose from ambient to about 10 MPa. Over this range, the viscosity gF ranges from 15.3 to 16.1 lPa s; since the change is less than 4%, it is treated as constant, gF ¼ 15:9 lPa s. The calculated density data are shown in Fig. 5, along with a

With this condition, the solution of Eq. (15) becomes Z 1 dX hPf ðhÞi ¼ PA ðhÞ
/ðh
h0 Þ 0 dh0 dh 0
 A
/ðhÞ: ð26Þ 8b

ez ¼ hef i


PA : 3Kp

ð27Þ

hPF i
PA : 3Kp

44

0. 5

42 0. 4

ρF ( g / cm3 )

T

40

0. 3

°

38 0. 2 36 0. 1

Using Eq. (4), this becomes ez ¼ hes i þ

46

34

ρF

ð28Þ

Eqs. (22), (26), and (28) constitute the central result of this paper. Experimental data for the axial strain of the sample are fit to these equation with sD as the only free parameter; using an indepen-

T (C)

According to Eqs. (25) and (32) of [4], the axial strain of the sample is given by

0. 6

32

0 0

2

4

6

8

10

12

PF ( MPa )

Fig. 5. The symbols are density values for CO2 calculated from equations given in [14], taking account of the pressure and temperature in the autoclave.

40

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

polynomial fit. For the calculations, piecewise polynomials were used over pressure intervals of 0–2, 2–4, 4–6, 6–8, 8–9, and 9–10.3 MPa. The driving term in Eq. (26) is dX=dh, which can be written in terms of the fluid pressure, as follows: 1 dX 1 dqF dhPF i dhPF i ¼ ¼ CF ; ð30Þ ð1
qÞKp dh qF dhPF i dh dh where CF is the compressibility of the fluid. For an ideal fluid, CF ¼ 1=PF , and Fig. 6 shows that this relationship holds true for CO2 at pressures up to about 2 MPa. If the fluid is ideal, then nMw PF Mw ¼ ; ð31Þ qF ¼ V Rg T where n ¼ number of moles, Mw ¼ molecular weight, V ¼ volume, and Rg ¼ ideal gas constant. In this case, Eq. (11) can be written as
 hPF ðhÞi XðhÞ ¼ ð1
qÞKp ln : ð32Þ hPF ð0Þi Fits to all the data were repeated using Eq. (32) in place of Eq. (11).

5. Results The load relaxation obtained in the beambending experiment is shown in Fig. 7. The fit indicates that PoissonÕs ratio is 0.20, YoungÕs modulus is 1.87 MPa, so the bulk modulus is 1.0 MPa, and the permeability is D1 ¼ 13:5 nm2 . The acoustic velocity was 170 m/s and the bulk density of the sample was 207 kg/m3 , so the longitudinal modulus was c11 ¼ Kp =b ¼ 5:97 MPa. Since mp ¼ 0:2, then b ¼ 0:5, and Kp ¼ 3:0 MPa. The total data collection is shown in Fig. 8. The fits were performed separately on each pressure jump, using the procedure explained in Appendix C. The pressure steps were relatively small, so the viscosity of the fluid, gF , was treated as constant within each data set, as were the parameters a and sD . The quality of the fits, illustrated in Fig. 9, is generally excellent for all pressures up to about 6 MPa. At pressures approaching the critical point of CO2 (7.5 MPa), the calculated compressibility is unreliable, so those data were not satisfactorily fit. The fits (not shown) were equally good when the fluid properties were assumed to be ideal; that

4.0

1.00 3.5

0.95

2.5

W(t) / W(0)

-1

C ( MPa )

3.0

F

2.0

1.5

0.90

0.85 1.0

Data a Fit

0.5

0.80

0.0 0

0.5

1

1.5

2

2.5

3

3.5

4

-1

1 / PF ( MPa )

0.75 0. 1

1

10

100

1000

t ( s)

Fig. 6. The symbols are compressibility values for CO2 calculated from equations given in [14], taking account of the pressure PF and temperature in the autoclave; the values are erratic near the critical pressure, Pcrit ¼ 7:36 MPa (1=Pcrit ¼ 0:14). The dashed line represents the behavior of an ideal gas, for which CF ¼ 1=PF .

Fig. 7. Force, W ðtÞ, normalized by initial value, measured in beam-bending experiment on wet gel exchange into pure ethanol. The data are fitted to the theoretical curve [11] to obtain the hydrodynamic relaxation time, s, which is used to calculate the permeability of the wet gel.

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

PA

10

εz

2

8

1. 5

6

1

4

0. 5

2

0

Axial Strain ( % )

PA ( MPa )

is, when X was represented by Eq. (32) in place of Eq. (11).

2. 5

12

6. Discussion The permeability of the aerogel can be calculated from the fitted values of sD using Eq. (14) together with the known properties of the fluid and the results of acoustic measurements on the aerogel (Kp ¼ 3:0 MPa and m ¼ 0:2). The permeability is predicted to depend on the pressure in the fluid according to [22]

-0.5

0 0

500

1000

1500

2000

2500

3000

t (s)

Fig. 8. Total data collection showing autoclave pressure jumps and resulting axial strain of aerogel.

PA PA Smooth

P

ε (meas)

ε z (calc)

P Smooth

ε (calc)

ε (calc)

z

A

z

A

s

F

2.00

0.80

0.360

0.20

1.90

0.60

0.340

0.10

1.80

0.40

0.320

0.00

1.70

0.20

0.300

-0.10

1.60

0.00

-0.20

1.50

75

80

85

(a)

90

A

P

A

z

0.280

( MPa )

0.30

95

-0.20 520

530

540

(b)

t (s)

εz (%)

0.380

ε (%)

P ( MPa )

ε z (meas) εs (calc)

F

41

550

560

570

t (s)

P

ε (meas)

P Smooth

ε (calc)

ε (calc)

z

A

z

A

s

F

5.30

1.30

1.20

5.25

z

ε (%)

1.00

A

P ( MPa )

1.10 5.20

5.15 0.90 5.10

5.05 1665

(c)

0.80

1670

1675

1680

1685

0.70 1690

t (s)

Fig. 9. Fit to pressure jump at average pressure of about (a) 0.36 MPa; (b) 1.8 MPa (Fits to multiple jumps were not included in the data in Fig. 10 or 11; each jump was fit separately.) and (c) 5.2 MPa.

42

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47


D ¼ D1

 k 1þ ; PF

ð33Þ

where k 4ck ¼ : PF rc

ð34Þ

This equation is based on a simplified analysis of diffusion in a cylindrical capillary tube with radius rc . The average distance from the wall at which the last intermolecular collision occurs is ck, where k is the mean free path of the gas; it is expected that c  1. The mean free path is related to the diameter of the molecule, r, by [24] kB T k ¼ pffiffiffi ; p 2r2 PF

ð35Þ

where kB ¼ BoltzmannÕs constant, so 4ckB T k ¼ pffiffiffi : p 2r2 rc

ð36Þ

The dashed line in Fig. 10 is a fit in the form of Eq. (33); the pressure is adjusted by atmospheric pressure, Patm ¼ 0:1 MPa, because the autoclave records pressure relative to Patm . For all the data in the range 0:36 6 PA ðMPaÞ 6 7:6, the fit yields D1 ¼ 8:90  0:17 nm2 and k ¼ 2:76  0:07 MPa.

The molecular diameter of carbon dioxide obtained from viscosity measurements [23] is 0.33 nm, so Eq. (34) indicates that the mean pore size of the aerogel is rc  13 nm. Another estimate of the pore radius, rp , can be obtained from the liquid permeability using the Kozeny equation [25] D1 ¼

ð1
qÞrp2 ; 4jw

ð37Þ

where jw is called the Kozeny constant, but is actually a weak function of relative density [26] that can be approximated by [25] jw ¼ 1:0 þ 6:05q1=2
8:60q þ 6:56q3=2 :

ð38Þ

Given D1 ¼ 8:9 nm2 and q ¼ 0:1, Eq. (37) indicates that rp ¼ 9:5 nm, which is about 30% smaller than rc . Data in the pressure range below 2 MPa were fit assuming ideal behavior for the fluid, leading to the results shown in Fig. 11. For this range, the fit yields D1 ¼ 7:3  1 nm2 and k ¼ 3:5  0:6 MPa, so Eq. (34) indicates that the mean pore size of the aerogel is rc  9–12 nm; from D1 we estimate rp  8–9 nm. Thus, the assumption of ideality yields reasonably accurate results.

100 100

80 80

60 2

D ( nm )

2

D ( nm )

60

40

40

20

20

0

0

0

0.5

1

1.5

2

2.5

3

3.5

-1

1 / ( P + Patm ) ( MPa ) F

Fig. 10. Plot of permeability versus absolute fluid pressure; dashed curve is fit to Eq. (33). Permeability data extracted using density data from [14] for carbon dioxide.

0

0.5

1

1.5 1/(P +P A

2

2.5

3

3.5

-1

atm

) ( MPa )

Fig. 11. Plot of permeability versus absolute fluid pressure; dashed curve is fit to Eq. (33). The permeability data were extracted assuming ideal behavior for carbon dioxide.

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

Before supercritical drying, the permeability of the wet gel was measured using the beam-bending method [11]; the liquid permeability, which corresponds to D1 , was found to be 13.5 nm2 . The relative density of the wet gel was q ¼ 0:068 (so jw  2:1), so Eqs. (37) and (38) indicate that the pore radius in the wet gel is rp  11 nm. After supercritical drying the gel had contracted so that its density increased to q ¼ 0:10 (so jw  2:3). Previous studies [6,25,27,28] have shown that the pore size of gels varies in proportion to the pore volume, Vp , which is given by
 1 1 Vp ¼
1 ; ð39Þ qs q where qs is the skeletal density; for silica aerogels, qs  2:0 g/cm3 . Given that the pore volumes of the gel and aerogel are 6.85 and 4.5 cm3 /g, respectively, the pore radius in the aerogel is expected to be rp  7:3 nm. In view of the large adjustment in the pore volume, this compares favorably with the value (9.5 nm) found using D1 from Fig. 10. Eq. (37) indicates that the permeability of a wet gel with the structure of the aerogel (rp ¼ 7:3 nm, q ¼ 0:1) should be D1  5:3 nm2 , which is reasonably close to the value (8.9 nm2 ) found from Fig. 10. The BET surface area of the aerogel was found to be S ¼ 920 m2 /g. Using the pore volume of Vp ¼ 4:5 cm3 /g, the hydraulic radius is found to be rh ¼ 2Vp =S  9:8 nm. Stumpf et al. [5] obtained a formula relating k to the viscosity of the fluid: sffiffiffiffiffiffiffiffiffiffiffiffiffi 32cgF 2pRg T k¼ ; ð40Þ Mw 3prc which yields rc  12 nm, using the data from Fig. 10, and rc  8–11 nm, using the data from Fig. 11. The value of rc obtained from Eq. (40) that was reported in [5] was found to be erroneous [7], 1

1 Regarding [5]: The increase of the diffusion coefficient D with gas pressure shown in Fig. 3 is not due to viscous flow as claimed in the text, but is an artifact caused by a variation of V1 , V2 and thus / as a function of pressure. As the diffusion coefficient D is normalized to / ¼ 1 it seemingly becomes also pressure dependent. The data for pave ! 0 however are reliable. Using the y-intercept of DHe a pore radius of 10 nm can be calculated.

43

owing to deformation of the apparatus; however, the values extrapolated to zero pressure were valid, and those values yielded a pore radius that was about 70% of the hydraulic radius and 40% of the half-chord length from SAXS. Similarly, in [6] the pore size found from gas transport was about 40% less than the chord length found from SAXS. Other studies have found a closer agreement between the pore sizes inferred from liquid permeability and nitrogen desorption [25], or from SAXS and nitrogen desorption [29]. Unfortunately, the aerogel used in the present study was too compliant for measurement of the pore size by nitrogen desorption; it was severely compressed by capillary pressure during condensation of liquid nitrogen [30,31]. Only about 0.6 cm3 /g out of the total pore volume of 4.5 cm3 /g were detected, so a large correction would be required to obtain the actual pore size [25,29]. Nevertheless, the BET surface area is expected to be accurate, since it is obtained from adsorption at low pressure.

7. Conclusions By measuring the linear strain of an aerogel during depressurization of the autoclave, it is possible to determine the permeability as a function of the fluid pressure. A simple modification of the autoclave (viz., installation of an LVDT in the pressure chamber) allows this measurement to be done routinely during supercritical drying. The analysis involves fitting Eqs. (12), (22), (26), and (28) to the strain data with a single free parameter, the characteristic time, sD . We have performed this calculation using data obtained for a silica gel with a relative density of 0.10, using the true properties of fluid CO2 or treating the fluid as an ideal gas. The fits are excellent and the results of the two calculations are surprisingly similar. The dependence of permeability on pressure obeys KlinkenbergÕs model; moreover, the permeability obtained by extrapolating the data for the fluid to infinite pressure agrees reasonably well with the permeability measured in liquid. Comparable estimates of the mean pore radius are obtained, based on the hydraulic radius (9.8 nm), liquid permeability (7.3– 9.5 nm), and gas diffusion (12–13 nm).

44

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

Acknowledgements The authors are indebted to Dr Gudrun Reichenauer for several helpful suggestions. This work was supported by DOE contract DEFG 0297ER45642. Appendix A. Isothermal versus adiabatic compressibility There are two extreme possibilities as to which modulus is appropriate for the fluid. If the pressure changes during the experiment are slow compared to the thermal relaxation time of the sample, the isothermal bulk modulus of the fluid has to be used. On the other hand, if the processes in question are fast enough to ensure thermal isolation of the sample – no heat is exchanged between the sample and the surrounding fluid on the time scale of the experiment – then a composite adiabatic modulus is appropriate. This is a composite modulus, instead of the pure adiabatic modulus of the fluid, because the fluid locally exchanges heat with the skeleton (the thermal relaxation time of heat transfer between the pore fluid and the skeleton is very short, of the order of ns). In this case, the bulk modulus of the fluid can be calculated as follows. The general definition of the bulk modulus is dp K ¼
V : ðA:1Þ dV An adiabatic process implies dQ ¼ dU
dW ¼ 0, that is Cv dT þ p dV ¼ 0;

ðA:2Þ

where Cv is the combined heat capacity of a certain volume containing skeleton and fluid. It is given by Cv ¼ mf Cvf þ mg Cvg ;

ðA:3Þ

where Cvf and Cvg are the specific heats of the fluid and the skeleton, respectively and mf and mg are the corresponding masses in the volume. From (A.2) we can deduce the connection between temperature and pressure changes for adiabatic processes p dT ¼
dV ; ðA:4Þ Cv

which allows us to write the adiabatic bulk modulus as   
dp  dp  p dp  ¼
V
Kad ¼
V dV ad dV T Cv dV v   dp  p dp  ¼ q  þ : ðA:5Þ dq T ð1
qÞqf Cvf þ qg Cvg dT v Here we have introduced the mass density of the fluid qf and the apparent density of the aerogel, qg . This is a general expression that can be used to calculate the composite adiabatic bulk modulus of the fluid in gel pores from the equation of state of the fluid, which enters through the derivatives of p and the fluid density. Note that this is also the modulus that determines the contribution of the fluid to the sound velocity. Using the ideal gas equation of state, Eq. (A.5) becomes identical to the modulus derived for that purpose earlier [32]. The modulus in Eq. (A.5) is position-dependent within the gel, but it is reasonable to neglect this as long as the temperature and pressure variation within the gel is small. For the numerical fitting in [9], the pressure and temperature measured in the autoclave were used to calculate the adiabatic fluid bulk modulus. If the thermal relaxation time is of the same magnitude as the hydrodynamic relaxation time, then the thermal diffusion problem must be introduced into the analysis. This is not necessary here, since the thermal relaxation is about a factor of 10 slower than the hydrodynamic relaxation for aerogels. Appendix B. Sum of Bessel functions A function yðu; tÞ can be expanded in a series of Bessel functions as 1 X yðu; tÞ ¼ an ðtÞJ0 ðBn uÞ; ðB:1Þ n¼1

where Bn is a root of J0 ðBn Þ ¼ 0:

ðB:2Þ

The Bessel transform of y is defined by [18] Z 1 yðu; tÞJ0 ðBn uÞu du: ðB:3Þ y~n ðtÞ ¼ 0

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

Substituting Eq. (B.1) into Eq. (B.3) leads to an expression for the coefficients an ðtÞ: an ðtÞ ¼

2~ yn ðtÞ J1 ðBn Þ

2

ðB:4Þ

:

With Eq. (B.4), Eq. (B.1) becomes 1 X 2~ yn ðtÞJ0 ðBn uÞ : yðu; tÞ ¼ 2 J1 ðBn Þ n¼1

ðB:5Þ

With the expansion in Eq. (B.5), the following integral leads to a simple result: Z 1 1 X 2~ yn ðtÞ : ðB:6Þ yðu; tÞu du ¼ B J n 1 ðBn Þ 0 n¼1

slowly changing pressure, then a jump on the order of 0.1–0.3 MPa, followed by another period of constant or slowly changing pressure. The total duration of each set is about 20 s. Since the data collection rate was relatively high, it is reasonable to assume that the average fluid pressure, hPF i, varies linearly within each time interval hPF ðhÞi ¼ ak þ bk ðh
hk Þ;

It is known that Z 1 I1 ðxÞ : I0 ðxuÞu du ¼ x 0

ðB:7Þ

n¼1

x2

1 I1 ðxÞ : ¼ 2 þ Bn 2xI0 ðxÞ

ðC:1Þ

ak ¼ hPF ðhk Þi

ðC:2Þ

and bk ¼

hPF ðhkþ1 Þi
hPF ðhk Þi : hkþ1
hk

ðC:3Þ

Then Eq. (30) leads to ðB:8Þ

Using Eq. (B.8) to represent the left side of Eq. (B.6), and substituting Eq. (B.7) into the right side of Eq. (B.6), we obtain 1 I1 ðxÞ X 2I0 ðxÞ ¼ ðB:9Þ 2 þ B2 x x n n¼1 or 1 X

hkþ1 P h P hk ;

where we define the constants

Now, if yðu; tÞ ¼ I0 ðxuÞ, then Eq. (B.3) yields Bn J1 ðBn ÞI0 ðxÞ y~n ðtÞ ¼ : x2 þ B2n

45

dX ¼ ck ðakþ1
ak Þ  wk ; dh

hkþ1 P h P hk ;

where ck ¼

ð1
qÞKp CF ðak Þ : hkþ1
hk

ðC:5Þ

With this approximation, the integral in Eq. (25) becomes Z hn Z hkþ1 n
1 X dX /ðhn
h0 Þ 0 dh0 ¼ wk /ðhn
h0 Þ dh0 dh 0 h k k¼1

ðB:10Þ ¼

n
1 X

wk ½Wðhn
hk Þ

k¼1

This result is used in Eq. (21).


Wðhn
hkþ1 Þ; Appendix C. Numerical evaluation The measured values of pressure in the autoclave were too noisy to use for numerical differentiation, so they were smoothed with a cubic spline [21]. The smooth and raw data are shown in the figures, which show that the procedure works well. The spacing between the data points is about 0.12 s, so no interpolation was required. In some sets there were larger intervals, because of quirks in the data logging software, and those sets were discarded. A Ôdata setÕ consists of a constant or

ðC:4Þ

ðC:6Þ

where WðhÞ ¼

Z

h

/ðh0 Þ dh0 :

ðC:7Þ

0

This function is given to better than 1% accuracy by  
 pffiffiffi h0:5
h0:7285 1 1
exp
4 h WðhÞ  : 4 1
h0:2722 ðC:8Þ The exact function and the approximation are shown in Fig. 4.

46

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47

With Eq. (C.6), Eq. (26) becomes an ¼ zn
cn
1 ðan
an
1 ÞWðhn
hn
1 Þ; where




zn ¼ PA ðhÞ


ðC:9Þ

 A /ðhÞ
w1 Wðhn Þ 8b

þ wn
2 Wðhn
hn
1 Þ


n
2 X

ðwk
wk
1 ÞWðhn
hk Þ:

ðC:10Þ

k¼2

so

Since an appears on both sides of Eq. (C.9), we must rearrange it to obtain zn þ cn
1 an
1 Wðhn
hn
1 Þ an ¼ : 1 þ cn
1 Wðhn
hn
1 Þ

ðC:11Þ

This equation allows the fluid pressure to be evaluated at time hn using values obtained in previous time steps. The sum in zn exists only when n > 2. If we number the intervals such that h1 ¼ 0, then
 A a1 ¼ PA ð0Þ
: ðC:12Þ 8b In the next time step

A PA ðh2 Þ
8b /ðh2 Þ þ c1 a1 Wðh2 Þ : a2 ¼ 1 þ c1 Wðh2 Þ

tained, and the average of the two was used for the calculation. To evaluate the baseline strain, we assume that the steady-state conditions exist at the start of the data set, so Eq. (C.12) applies. Then, using Eqs. (12) and (25), the initial strain is found from Eq. (28) to be

 1 þ 3Kp a A ðC:15Þ ez ð0Þ ¼
0 þ aPA ð0Þ
3Kp 8b

a¼

A ez ð0Þ

0 þ ð8bÞð3K pÞ A PA ð0Þ
8b

:

ðC:16Þ

Steady-state is assumed to exist at the end of each data set, so another value of a is determined from the final strain value, and the average of the two values of a is used for the fitting. All fits were performed using the experimental values (see Section 6) of mp ¼ 0:2, b ¼ 0:5, and Kp ¼ 3:0 MPa.

References

ðC:13Þ

All successive pressures are found from Eq. (C.11). The strains are then calculated from Eq. (28). To test the ideal gas approximation, CF is replaced by hPF ðhÞi in Eq. (C.5). To calculate the correct value of a for evaluation of Eq. (12), it is necessary to take account of the portion of the initial strain that results from the steady-state pressure. For that purpose, leastsquares lines were fit through the first five points of each data set to find the change in autoclave pressure with time. The slope of this line, dPA =dt, was used to estimate A   dX  dPA  A¼  ð1
qÞK C ½P ð0Þs : p F A D dh h!0 dt t!0 ðC:14Þ Similarly, the slope was found for the last five points in the set and another value of A was ob-

[1] T. Heinrich, U. Klett, J. Fricke, J. Porous Mater. 1 (1) (1995) 7. [2] S.S. Prakash, C.J. Brinker, A.J. Hurd, S.M. Rao, Nature 374 (1995) 439. [3] G.W. Scherer, J. Non-Cryst. Solids 145 (1992) 33. [4] G.W. Scherer, J. Sol–Gel Sci. Technol. 3 (1994) 127. [5] C. Stumpf, K.V. G€assler, G. Reichenauer, J. Fricke, J. Non-Cryst. Solids 145 (1992) 180. [6] G. Reichenauer, C. Stumpf, J. Fricke, J. Non-Cryst. Solids 186 (1995) 334. [7] G. Reichenauer, private communication. [8] G. Reichenauer, J. Fricke, in: J. Drake (Ed.), Dynamics in Small Confining Systems III, Materials Research Society, Pittsburgh, PA, 1996, p. 345. [9] J. Gross, Heidelberg paper. [10] C.J. Brinker, K.D. Keefer, D.W. Schaefer, R.A. Assink, B.D. Kay, C.S. Ashley, J. Non-Cryst. Solids 63 (1984) 45. [11] G.W. Scherer, J. Non-Cryst. Solids 142 (1&2) (1992) 18. [12] G.W. Scherer, J. Am. Ceram. Soc. 83 (9) (2000) 2231. [13] G.W. Scherer, J. Sol–Gel Sci. Technol. 1 (1994) 169. [14] M.P. Vukalovich, V.V. Altunin, Thermophysical Properties of Carbon Dioxide, ColletÕs, London, 1968. [15] M.A. Biot, J. Appl. Phys. 12 (1941) 155. [16] D.J.C. Yates, Trans. Br. Ceram. Soc. 54 (1955) 272. [17] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, 1986.

J. Gross, G.W. Scherer / Journal of Non-Crystalline Solids 325 (2003) 34–47 [18] F.B. Hildebrand, Advanced Calculus for Applications, Prentice-Hall, Englewood Cliffs, NJ, 1962. [19] Wolfram Research, Inc., 100 Trade Center Drive, Champaign, IL 61820-7237 USA. [20] D.M. Olsson, J. Qual. Technol. 6 (1974) 53. [21] IMSL Math/Library routine CSSMH, copyright Visual Numerics, Inc., 9990 Richmond Avenue, Suite 400, Houston, Texas 77042-4548, USA. [22] L.J. Klinkenberg, Drill. Proc. API (1941) 200. [23] R.C. Weast, M.J. Astle (Eds.), CRC Handbook of Chemistry and Physics, 62nd Ed., CRC, Boca Raton, FL, 1981, p. F-172. [24] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, p. 20. [25] G.W. Scherer, J. Non-Cryst. Solids 215 (2&3) (1997) 155.

47

[26] G.W. Scherer, J. Sol–Gel Sci. Technol. 1 (1994) 285. [27] S. Wallace, C.J. Brinker, D.M. Smith, in: A.K. Cheetham, C.J. Brinker, M.L. Mecartney, C. Sanchez (Eds.), Better Ceramics Through Chemistry VI, Materials Research Society, Pittsburgh, PA, 1995, p. 241. [28] D.M. Smith, G.W. Scherer, J.M. Anderson, J. Non-Cryst. Solids 188 (1995) 191. [29] G. Reichenauer, G.W. Scherer, J. Non-Cryst. Solids 285 (2001) 167. [30] G.W. Scherer, D.M. Smith, D. Stein, J. Non-Cryst. Solids 186 (1995) 309. [31] G. Reichenauer, G.W. Scherer, J. Non-Cryst. Solids 277 (2000) 162. [32] J. Gross, J. Fricke, L.W. Hrubesh, J. Acoust. Soc. Am. 91 (4) (1992) 2004.