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Optimization of the rapid supercritical extraction process for aerogels
Journal of Non-Crystalline Solids 311 (2002) 259–272 www.elsevier.com/locate/jnoncrysol
Optimization of the rapid supercritical extraction process for aerogels George W. Scherer
a,*
, Joachim Gross a,1, Lawrence W. Hrubesh b, Paul R. Coronado b
a
b
Princeton University, CEOR/PMI, Eng. Quad. E-319, Princeton, NJ 08544, USA Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 USA Received 17 October 2001; received in revised form 28 February 2002
Abstract A partial differential equation is derived that describes the pressure developed in the pores of a gel during the rapid supercritical extraction process. A comparative analysis of the strains caused by syneresis and expansion of the fluid, respectively, suggests that the latter is the dominant effect for this process. Experimental results indicate that the rate of leakage from the mold is equal to the rate of volumetric expansion of the fluid, so this was used as the boundary condition for the calculation. An analytical solution is obtained for the strain produced in a purely elastic gel. The strain is found to develop most rapidly at high temperatures, where the thermal expansion of the fluid increases sharply. The model predicts a temperature dependent heating rate that can be used to avoid irreversible strains by compensating for the increase in thermal expansion coefficient. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The rapid supercritical extraction process (RSCE) permits extraordinarily rapid preparation of aerogels [1]. The wet gel is placed into a mold, which is then inserted into an autoclave. As the temperature of the autoclave rises, the temperature and pressure in the mold increase until the fluid begins to leak from the mold; deformation of the gel is largely prevented because it is constrained *
Corresponding author. Tel.: +1-609 258 5680; fax: +1-609 258 1563. E-mail address: [email protected] (G.W. Scherer). 1 Present address: Zentrum Fuer Medizinische Forschung, Waldhoernlestr. 22, Tuebingen 72072, Germany.
inside the mold, so the temperature can be raised beyond the critical point much faster than in conventional supercritical drying. Unfortunately, the resulting aerogels often exhibit spatially varying properties. The inhomogeneity of the aerogels is believed to result from pressure gradients created as the vapor is vented from the mold. Therefore, a mold was developed with a fritted plate on one end, so that the pressure would be released uniformly over a larger area. The purpose of this paper is to analyze the gradients in that type of mold, and to identify the parameters that must be controlled to minimize them. The analysis is an extension of earlier studies of stresses developed during thermal expansion of the pore liquid [2,3] and depressurization of the autoclave [4,5].
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 3 7 9 - 0
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2. Experimental procedure 2.1. Rapid supercritical extraction process The original RSCE mold was cylindrical with a diameter of 29.3 mm and length of 23.3 mm. The mold was filled with a sol (described in Section 2.3) and the end plates were bolted on. The mold was placed inside an autoclave that was filled with methanol; the pressure in the liquid was pumped to 2000 psi (13.8 MPa) then the assembly was heated. As the temperature rose, the pressure in the mold eventually forced the lid open slightly, so that fluid leaked out. Two experiments were performed to study the rate of leaking by capturing and weighing the effluent. In one case (experiment Para-A-35), the mold contained only methanol,
and in another (Para-A-41) it contained a sol of the type described below. A new cylindrical mold was made with the same dimensions and design, except that a steel frit was placed over one end. The samples were subjected to the temperature–pressure history shown in Fig. 1, where the pressure immediately ramped to about 2000 psi (13.8 MPa) before the temperature was raised. This experiment is identified as RSCE B-271. In both experiments, the accuracy is about 3% for the pressure and 2% for the temperature measurements. 2.2. Density profiles To ascertain the uniformity of the aerogels, the acoustic pulse velocity was measured on disks cut from the cylindrical gels. The disks were 30 mm in diameter and about 2 mm thick, as shown in Fig. 2. The acoustic velocity was measured with 180 kHz transducers, as explained in detail in Ref. [6]. The accuracy of the sound velocities is about 2%, or about 2–10 m/s. 2.3. Permeability and modulus
Fig. 1. Temperature and pressure in the fritted mold during RSCE experiment.
Rods of silica gel were made using the same recipe as for RSCE. Solution A was prepared by mixing 35.98 g of methanol (99.8þ% from Aldrich) with 30.06 g of tetramethoxysilane (TMOS,
Fig. 2. Schematic (a) and photo (b) of fritted mold for RSCE.
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99þ% from Aldrich). Solution C was made using 196.62 g methanol and 3.41 g NH4 OH, and solution B was prepared using 21.84 g H2 O, 36.14 g methanol and 1.02 g of solution C. Solution B was then poured into solution A and the stirring was continued for 10 more minutes. The homogeneous gel solution was poured into polystyrene molds of 7.9 mm internal diameter and the molds were capped with rubber septa, then the gels were cured in the incubator at 49 °C. One or two molds were taken out of the incubator after aging for 1, 2, 3, 4, 6, 8, and 12 days, respectively. The gel beams were transferred from the mold to the 20 ml test-tubes and were washed with methanol five times within 3 days. The gels were tested by beam-bending [7–9] using a 50 g load cell with methanol as the bath solution. The span for three-point bending was 82.2 mm and the deflection was set at 1.5 mm, which corresponds to 6 1% strain in the gel beams. Upon application of the deflection, a pressure gradient is created in the methanol in the pores, so the liquid begins to flow within the beam and to exchange with the surrounding bath. As the pore pressure equilibrates, the force required to sustain a constant deflection decreases; by fitting the force decay to the theoretical expression [8], the permeability, elastic modulus, and PoissonÕs ratio for the gel can be obtained.
3. Results 3.1. Beam-bending The gels are relatively compliant, so some initial loading data (for about 1 s) failed to be recorded
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due to vibration in the beam-bender and the small loading value (around 0.3 g force). The summary of the beam-bending results is shown in Table 1. The permeability is calculated from the hydrodynamic relaxation time by taking 0.544 mPa s as the viscosity of methanol. A typical plot of the hydrodynamic relaxation curve measured in the beam-bending experiment is shown in Fig. 3. The fit is reasonably good, but the load is small because of the compliance of the gel, and this is responsible for the scatter in the results reported in Table 1. The permeability of the gel is 32 10 nm2 and the YoungÕs modulus is 0.1 MPa; neither property shows systematic variation with age, although the modulus may be rising at 12 days of aging. As the gel is heated in the autoclave, the modulus will increase; measurements of aerogels made from this recipe indicate that E ¼ 2:1 MPa. However, since the volume of the gel does not change appreciably, the permeability (which depends on the porosity and pore size) is expected to remain nearly constant. 3.2. Density gradients in wet gels Before considering the shrinkage during RSCE, we will discuss the density gradients observed in gels prepared under ambient conditions. The refractive index was measured on wet gels using methods described elsewhere [10]. Wet gels show a region within a few millimeters of the outer surface where the density is higher than in the interior, even if the sols, gels and molds are maintained at room temperature. It was recently reported that the shrinkage of a gel is related to surface charges and solvent polarity [11], so shrinkage near the surface of the gel might result from contamination
Table 1 Summary of beam-bending data Aging (days)
Beam-bending Poisson ratio
Hydrodynamic s (s)
Shear modulus G (MPa)
Elastic modulus E (MPa)
Permeability D (nm2 )
Gel diameter d (mm)
1 2 3 4 6 8 12
0.169 0.142 0.105 0.274 0.127 0.147 0.138
5890 3770 6570 2940 3900 4580 1970
0.0436 0.0366 0.0357 0.0412 0.0447 0.0498 0.0695
0.102 0.0835 0.0789 0.105 0.101 0.114 0.158
21.6 43.6 27.6 30.9 35.3 25.0 41.9
7.86 7.81 7.76 7.79 7.79 7.72 7.62
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where gF is the viscosity of the pore fluid, D is the permeability of the network, HG is the longitudinal viscosity of the drained network, and b ¼ ð1 þ mÞ= ½3ð1 mÞ, where m is PoissonÕs ratio; In is a modified Bessel function of the first kind of order n. For a rigid gel (large HG ) or a highly permeable gel (large D), a is small and Eq. (1) reduces to lim e_ðrÞ ¼ 3e_s a!0
Fig. 3. Relaxation of load, W, during beam-bending experiment performed on gel aged 8 days; wVE represents viscoelastic part of relaxation.
of the pore liquid by corrosion of the mold; that is, ions released by corrosion of the mold could alter the surface charges on the gel network and thereby influence dilatation. However, chemical analysis of thin gels (which have maximal contact with vessel walls) by ICP has revealed only tens of ppm of contamination, which is not likely to cause appreciable increase in shrinkage rate. It is known that the rate of syneresis is affected by the size of the gel, because shrinkage requires escape of the liquid from the network [12,13]. Within a large gel, the syneresis rate is greater near the exterior surface, where the pore liquid escapes freely from the network. The pressure distribution in the pore liquid has been calculated for a cylindrical gel with a viscous or viscoelastic network [13]. For the viscous case, the volumetric strain rate in a cylindrical gel with radius r0 is abI0 ðar=r0 Þ e_ðrÞ ¼ 3e_s ; ð1Þ aI0 ðaÞ 2ð1 bÞI1 ðaÞ where e_s is the linear syneresis strain rate inherent to the network (which is equal to the observed rate only in very thin gels, where flow of the liquid does not inhibit contraction). The parameter a is defined by sffiffiffiffiffiffiffiffiffiffi gF r02 a¼ ; ð2Þ DHG
ð3Þ
so the volumetric strain rate approaches the inherent rate (the factor of 3 appears because e_s is a linear strain rate) when a is small. For a wide variety of silica and organic gels examined in our lab [14,15], we find that 0:15 6 m 6 0:25; if m ¼ 0:2, then b ¼ 0:5, and Eq. (1) becomes aI0 ðar=r0 Þ e_ðrÞ 1 ¼ : ð4Þ 2 aI0 ðaÞ I1 ðaÞ 3e_s This result is plotted in Fig. 4 for three values of a. When a is small, the contraction rate is uniform throughout the cylinder, but as a increases, the contraction is more severely inhibited deep within the body; therefore, the density of the exterior will be increased by syneresis compared to the interior. Based on the beam-bending data for the sample shown in Fig. 3, the value of a for a cylindrical gel with a bulk density of qb ¼ 0:1 g/cm3 and r0 ¼ 2 cm is expected to be on the order of 0.1, so the exterior is not expected to have a significantly higher shrinkage rate than the interior.
Fig. 4. Volumetric strain rate normalized by inherent syneresis rate versus radial position for several values of a. Shrinkage rate is greatest at exterior surface.
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272
The only temperature-dependent factor on the right side of Eq. (4) is a, which depends on the ratio gF =HG ; it seems likely that the activation energy for viscous flow of the network will be somewhat higher than for the pore liquid, in which case a will increase with temperature, and the density gradients will be more severe. However, even if our estimate of a is low, Fig. 4 indicates that the variation in shrinkage rate varies roughly parabolically with radius, whereas the observed density increase was pronounced only within a few millimeters of the surface. Therefore, it appears unlikely that syneresis makes a significant contribution to the overall density gradients seen in aerogels made by RSCE. 3.3. Density gradients in RSCE gels The spatial variation in acoustic velocity in an RSCE aerogel prepared in a conventional (unfritted) mold is illustrated in Fig. 5, which is taken from Ref. [6]. The gradients are most pronounced near the upper perimeter of the gel, near the place where the vapor leaks out of the mold. Away from the end, the velocity is almost constant, so the density is probably also nearly constant. There are strong radial gradients in temperature in the mold, but neither those gradients nor the effect of syneresis could account for the shape of the variation seen in Fig. 5. It must be the venting process itself that causes the greatest variation in density.
Fig. 5. Longitudinal sound velocity as function of position measured in an RSCE aerogel made in a conventional (unfritted) mold.
263
The longitudinal sound velocity, V, measured in an aerogel made in the fritted mold is shown in Fig. 6; the disk numbers correspond to the slices shown in Fig. 2. The velocity is related to the longitudinal modulus by H ¼ qb V 2 ;
ð5Þ
where H ¼ ð1 mÞE=½ð1 2mÞð1 þ mÞ, E is YoungÕs modulus of the gel network, m is PoissonÕs ratio, and qb is the bulk density. Extensive measurements of the elastic modulus of aerogels [16] indicate that the modulus varies with density according to a power law of the form m
H ¼ H0 ðqb =q0 Þ ;
ð6Þ
where H0 is the modulus when the density of the network is q0 , and the exponent typically lies in the range 2:5 6 m 6 4. From Eqs. (5) and (6), we find that the velocity is related to the density of the aerogel by qb ¼ ðqm0 =H0 Þ
1=ðm1Þ
V 2=ðm1Þ :
ð7Þ
Since m is typically near 3.5 for silica aerogels [16], V is nearly linearly proportional to the density of the gel. Fig. 6 shows that the density is much more uniform across the radius of the disk than for gels made from RSCE molds without a fritted end (Fig. 5). Nevertheless, disks 5 and 6 (near the frit) are significantly denser than the rest of the aerogel. PoissonÕs ratio for silica gels and aerogels is typically found to be m 0:2 [17], so H 1:1E;
Fig. 6. Longitudinal sound velocity measured in aerogel made in fritted mold shown in Fig. 2, subjected to the thermal cycle given in Fig. 1.
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since E0 2:1 MPa for this type of aerogel when q0 ¼ 100 kg/m3 , we estimate that H0 ¼ 2:3 MPa. Eq. (7) can be used to estimate the densities of the slices in the disks in Fig. 2: if m ¼ 3:5, then the density in disk 6, where V 230 m/s, is 140 kg/m3 and the density in disk 2, where V 150 m/s, is 99 kg/m3 . Thus, disks 1–4 have about the expected density for this recipe, while disks 5 and 6 have densities increased by 30–40%.
4. Analysis of RSCE 4.1. Elastic case The data in Figs. 5 and 6 indicate that the density gradients in the aerogel result primarily from compression of the network during venting, so we will begin by calculating the strain caused by that phenomenon alone. We will consider only the uniaxial flow that is expected to occur in a fritted mold, shown schematically in Fig. 7. The mold vents uniformly across the fritted surface at z ¼ L when the fluid pressure, PF ðz; tÞ, satisfies
The strains are ex ¼ ey ¼ ef þ
1 ½ð1 mÞrx mrz E
ð10Þ
and 1 ½rz 2mrx ; ð11Þ E where E and m are YoungÕs modulus and PoissonÕs ratio for the gel network. The free strain ef is the linear strain that would occur in the absence of stress; in a thermal expansion problem it is the product of the thermal expansion coefficient and the change in temperature. Let the expansion of the mold caused by thermal expansion and internal pressure be e0 , where ez ¼ ef þ
e0 ¼
hPF i þ aM DT ; RM
ð12Þ
ð8Þ
where RM is the rigidity of the mold (which depends on the modulus and dimensions of the mold), aM is the linear thermal expansion coefficient of the mold, and hPF i is the average pressure in the fluid inside the mold. Then ex ¼ ey ¼ e0 and rz ð13Þ ez ¼ 3bef þ ð1 3bÞe0 þ ; H
where P0 is the threshold pressure at which the mold begins to leak. To find the stresses in a gel, the first step is to solve the corresponding thermal stress problem [18]. In the present case, the flow is only along z, so there is symmetry in the x and y directions. The stresses are
where H is the longitudinal modulus of the gel network and b was defined previously. The axial strain is Z 1 L ez dz ¼ e0 : ð14Þ L 0
rx ¼ ry :
Substituting Eq. (13) into Eq. (14) leads to
PF ð1; tÞ P P0 ;
ð9Þ
rz ¼ 3Kðe0 hef iÞ
ð15Þ
and ez ¼ e0 þ 3bðef hef iÞ;
ð16Þ
where K ¼ bH is the bulk elastic modulus of the network and Z 1 L hef i ¼ ef dz: ð17Þ L 0 The volumetric strain is e ex þ ey þ ez ¼ 3e0 þ 3bðef hef iÞ:
ð18Þ
For a gel, the free strain rate is given by [2,18] Fig. 7. Cross-section of mold having length L and radius r0 ; dimensionless length is u ¼ z=L.
ef ¼ es þ aS DT þ
PF 3K
ð19Þ
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which includes contributions from syneresis (es ), thermal expansion of the gel network (aS ¼ thermal expansion of the solid), and pressure in the pore fluid. From Eqs. (16) and (19), the axial strain is ez ¼ e0 þ
PF hPF i H
ð20Þ
so the volumetric strain is e ¼ 3e0 þ
PF hPF i : H
ð21Þ
4.2. Continuity equation
ð22Þ
where qF is the density of the fluid, q is the volume fraction of solids in the network, and VF is the volume of fluid per gram of network; J is the flux of fluid, which is related to the pressure in the fluid by DarcyÕs law [19,20]: J ¼
D rPF ; gF
ð23Þ
where D is the permeability of the network and gF is the viscosity of the fluid. It has been shown that [2] e_ ¼ ð1 qÞ
V_F þ 3aS qT_ VF
ð24Þ
so Eq. (22) becomes q_ F þ
qF 1 r ðqF J Þ: ðe_ 3aS qT_ Þ ¼ 1q 1q
ð25Þ
To link Eqs. (21) and (25), we need the relation between qF and PF (i.e., the equation of state of the fluid). For the sake of simplicity we adopt the following linear equation (which is adequate for the liquid state), q_ F P_ F ¼ 3aF T_ ; qF KF
The general solution for the uniaxial case is obtained by solving Eqs. (21), (25) and (26), but the problem is highly non-linear, and can only be treated numerically. A useful approximation is obtained if we consider only the case where the quantity qF D=gF is weakly dependent on position, so that it can be removed from the divergence operator on the right side of Eq. (25). Then the continuity equation takes the form 1q 1 1 3 D o2 PF þ e_T ¼ hP_ F i ; P_ F KF H H RM gF oz2 ð27Þ where
The equation of continuity (or conservation of mass) is given by [4] 1 o 1 ðq VF Þ ¼ r ðqF J Þ; VF ot F 1q
265
ð26Þ
where KF is the bulk modulus (the reciprocal of the compressibility) of the fluid.
e_T ¼ 3½qaS þ ð1 qÞaF aM T_ :
ð28Þ
Finally, we introduce the hydrodynamic relaxation time, s, defined by gF L2 DH and the dimensionless time Z t dt h¼ 0 s
s¼
ð29Þ
ð30Þ
so that dt ¼ s dh. With these variables, Eq. (27) can be written as oPF ð1 qÞH ohPF i 3H oeT 1 1þ H KF oh RM oh oh o2 PF ¼ : ð31Þ ou2 Since the modulus of the gel is completely negligible compared to that of the mold, H RM , the quantity H =RM will be neglected. The same argument could be made for the quantity H =KF , but we will retain that term for the moment.
4.3. Boundary condition There is some threshold pressure, P0 , beyond which the mold begins to leak. To get an idea of the dependence of the leakage rate on pressure, we have examined data for the rate of release of pure methanol from an RSCE mold (experiment ParaA-35). The internal and external temperature and pressure were measured and the escaping methanol
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was collected, with the results shown in Fig. 8. Segment H of the pressure curve shows the rapid increase in pressure as the methanol (which initially fills the mold completely) is heated. The pressure in the liquid in that stage is given by
nearly incompressible, the amount of fluid that must escape is the difference in volume between the contents of the mold and the mold itself,
PF ¼ 3KF ðaF aM ÞDT ;
where VM is the volume of the mold. Therefore, the mass of methanol collected is expected to be
ð32Þ
where aF and aM are the thermal expansion coefficients of the fluid and the metal mold, respectively. For methanol, aF 4:3 104 =°C and KF ¼ 7:7 108 Pa; for 304 stainless steel, aM ¼ 1:7 105 =°C, so Eq. (32) predicts dPF =dT
0:95 MPa=°C ¼ 138 psi/°C, whereas the measured slope is 123 psi=°C ¼ 0:85 MPa=°C. This is reasonable agreement in view of the existence of temperature gradients and, more importantly, the probable decrease in the bulk modulus of liquid methanol at high pressure. Methanol begins to be collected at an appreciable rate after the pressure reaches 2150 psi, and the rate of collection increases linearly with temperature while the pressure remains nearly constant; near the critical point (240 °C), the pressure and leakage rate jump. The mass of fluid escaping is dMF dV ; ¼ qF dt dt
ð33Þ
where V is the volume of fluid escaping. As the temperature increases, the volumetric expansion of the contents of the mold is 3aF DT ; since the fluid is
dV dT ¼ VM ð3aF 3aM Þ ; dt dt
ð34Þ
dMF dT ; ¼ qF VM ð3aF 3aM Þ dt dt
ð35aÞ
dMF dT ;
3qF VM aF dt dt
ð35bÞ
where the approximation is justified by the small expansion of steel compared to the fluid. The thermal expansion coefficient is defined by aF ¼
1 dqF 3qF dT
ð36Þ
so Eq. (35b) reduces to dMF dq dT
VM F dt dT dt
ð37Þ
or dMF dq
VM F : dT dT
ð38Þ
For methanol [3], the density along the saturation line is given approximately by 0:360 513 T ðKÞ qF ðg=cm3 Þ ¼ 0:2722 þ ð39Þ 1359 so dqF 0:027
: dT ð513 T ðKÞÞ0:64
Fig. 8. Pressure in mold (reactor) and weight of methanol condensed at exit. Arrows show sequence of cycle; portion labeled H represents pressure increase as liquid methanol is heated before any liquid escapes from mold.
ð40Þ
The measured rate of mass escaping is obtained by smoothing, then differentiating the curve in Fig. 8. Using those data for dMF =dT along with Eq. (40), Eq. (38) predicts that the quantity shown in Fig. 9 should be equal to VM . We see that Eq. (38) is obeyed from ambient temperatures to about 200 °C; the constant value of 12 cm3 agrees closely with the volume of the mold (2.5 cm dia 2:5 cm thick ¼ 12:3 cm3 ). This result indicates that the fluid in the mold is essentially incompressible, so that the rate of escape from the mold equals the rate of expansion, once the pressure is
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272
Fig. 9. Test of Eq. (38) using data from same experiment as Fig. 8.
sufficient to force open a leak. Near the critical point, the fluid expands more rapidly than it can escape (causing the downward curvature in Fig. 9), so the pressure rises. A similar experiment was performed (experiment Para-A-41) where the mold contained a sol chosen to yield a gel with a density of 0.1 g/cm3 . As shown in Fig. 10, the pressure in the mold jumps by 75% when gelation occurs, but there is no corresponding change in the rate of collection of fluid. The increase in pressure is a result of the increase in viscosity upon gelation. Since the sol is mostly methanol, Eq. (38) should still be obeyed in this case, and Fig. 11 shows that it is; the constant value is again 12 cm3 . There is a jump in Fig. 11 at 75 °C, when gelation occurs, but thereafter the plot remains flat almost all the way to the critical temperature. These results indicate that the appropriate boundary condition for calculating the stress in the gel during RSCE is to set the flux out of the mold, J, equal to the volumetric expansion of the gel: J jz¼L ¼
1 dV ¼ A dt
3VM dT ½ð1 qÞaF þ qaS aM ; A dt ð41Þ
where A is the area of the leak, and the quantity in brackets includes the expansion of the fluid, the
267
Fig. 10. Pressure in mold (reactor) and weight of fluid condensed at exit. Arrows show sequence of cycle. Label L indicates the onset of leakage and label G indicates the point at which gelation occurred.
Fig. 11. Test of Eq. (38) using data from same experiment as Fig. 10, where mold contains a gelling solution.
solid (i.e., the gel network), and the mold. Using Eq. (23), the boundary condition becomes D dPF 3VM dT : ¼ ½ð1 qÞaF þ qaS aM g dz dt A F
z¼L
ð42Þ 4.4. Elastic solution We seek the solution of Eq. (31) subject to the boundary conditions given by Eq. (42) and D dPF ¼0 ð43Þ g dz F
z¼0
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which indicates that there is no flux across the bottom end of the mold. If we choose to set time to zero at the moment when the leak begins, the initial condition is PF ðz; 0Þ ¼ P0 :
ð44Þ
The solution is PF ðu; hÞ ¼ P0 þ H
Z
h
Xðu; h h0 Þ 0
oeT 0 dh ; oh0
ð45Þ
where the thermal strain is eT ¼ 3½ð1 qÞaF þ qaS aM ðT TR Þ;
ð46Þ
where TR is room temperature. The relaxation function that appears in Eq. (45) is Xðu; hÞ 2
1 X
n
ð1Þ cosðnpuÞ expðn2 p2 hÞ:
ð51Þ
For methanol, we can use Eq. (39) for qF , and find aF from Eq. (36). Data are available [21] for the viscosity of methanol in the range of temperature and pressure used in this experiment, and they are well represented by 3:92 44:6 gF ðPa sÞ ¼ : ð52Þ T ðKÞ The resulting values for gF aF are shown in Fig. 12. Since this quantity rises rapidly as the critical point approaches, Eqs. (49) and (50) indicate that the strain in the gel will rise rapidly in that region. Suppose that instead of using a constant heating rate, we let the rate vary as
ð47Þ
oT c ¼ ; ot aF gF
ð48Þ
where c is a constant. In this case, Eq. (50) indicates that oeT =oh is roughly constant, so Eq. (49) becomes 1 2 oeT X ð1Þn cosðnpuÞ ez 2 ½1 expðn2 p2 hÞ: p oh n¼1 n2
n¼1
The average pressure in the mold is Z 1 hPF i ¼ PF ðu; hÞ du ¼ P0
oeT / gF aF : oh
0
so the average pressure remains constant, in keeping with Fig. 10. The axial strain in the gel network is found from Eqs. (20) and (45); neglecting the thermal expansion of the mold (e0 ), the result is Z h oeT ez ¼ Xðu; h h0 Þ 0 dh0 : ð49Þ oh 0
ð53Þ
ð54Þ
If the duration of the experiment is long compared to the hydrodynamic relaxation time of the gel, we can assume that t s, so h 1, and Eq. (54) reduces to
4.5. Effect of heating rate on deformation The principle problem with RSCE is the deformation of the final aerogel, so we are most interested in the implications of Eq. (49). The strain in the gel is related to the quantity oeT oeT gF L2 oT ¼s ¼ ½3ð1 qÞaF þ 3qaS 3aM oh ot DH ot 2 gF L oT ð3aF Þ ; ð50Þ
DH ot where the approximation is justified by the relatively small values of aS and aM . If the heating rate is constant, then
Fig. 12. Product of viscosity and thermal expansion coefficient for methanol along the saturated liquid/vapor line.
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272 1 2 oeT X ð1Þn cosðnpuÞ p2 oh n¼1 n2 1 oeT 2 1 u
: 2 oh 3
269
ez ðu; hÞ
ð55Þ
The maximum strain occurs at the end where the leak is occurring (u ¼ 1): ez ð1; hÞ
1 oeT oT
aF s ; 3 oh ot
h 1;
ð56Þ
where the second equality follows from Eq. (50). This indicates that the strain in the network can be kept to a safe level by choosing the heating rate such that a safe constant value of oeT =oh is preserved. This analysis indicates that the deformation of the aerogel is minimized by those factors that decrease s. The properties of the gel that are most important are the permeability (D) and elastic modulus (H), both of which increase with aging and thereby reduce s. If the thermal cycle included a hold above the gel point and before the onset of leaking, it would serve to coarsen the gel, raising D and H. 4.6. Simulation of the experiment We want to evaluate Eq. (49) for the thermal cycle shown in Fig. 1, which requires that we know the viscosity and thermal expansion coefficient of the fluid, and the modulus and permeability of the gel. The viscosity is given by Eq. (52) and the density of the fluid can be calculated from data for the compressibility factor, f, defined by f¼
PF Vm ; Rg T
Fig. 13. Compressibility factor for methanol at 141 bars (2080 psi, 14.3 MPa).
qF ¼
M w M w PF ; ¼ Vm fRg T
ð59Þ
where Mw is the molecular weight (for methanol, Mw ¼ 32). The density calculated in this way is accurately fit by aF ¼ 2:047 103 5:0142 105 T þ 9:0238 107 T 2 6:5805 109 T 3 þ 1:6248 1011 T 4 :
ð60Þ
The properties obtained from Eqs. (52) and (60) are shown in Fig. 14. The permeability of the gel can be assumed to remain constant near the measured value of 32 nm2 , but the elastic modulus changes from 0.1 MPa in the initial wet gel to 2.1 MPa in the aerogel, and the evolution of that property in the autoclave is not known. For the following
ð57Þ
where Vm is the molar volume and Rg is the gas constant; f ¼ 1 for an ideal gas. Data are available [22] for f in the range of temperature and pressure used in the RSCE experiment, and a good fit is provided by f ¼ 0:5521 1:6878 104 T ð°CÞ þ 0:3107 tanhð0:0203T ð°CÞ 6:391Þ;
ð58Þ
where T is in °C (in this equation only); the curve is shown in Fig. 13. The density of the fluid is
Fig. 14. Estimated thermal expansion coefficient and viscosity for the fluid in the aerogel during the thermal history shown in Fig. 1.
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simulations, we will assume that the modulus is constant and equal either to 0.1 or 2.1 MPa, in order to set bounds on the behavior. The axial strain, ez , is obtained by numerical solution of Eq. (49), using the temperature and pressure history from Fig. 1, along with the physical properties determined above. Since the strains in the lateral directions are constrained by the mold, the axial and volumetric strains are equal, and the density of the gel is related to the axial strain by q ¼ q0 expðez Þ:
ð61Þ
Fig. 15 shows the density as a function of temperature at the positions corresponding to the disks in Fig. 2, when E ¼ 0:1 MPa. The calculations in Fig. 15 indicate that disks 5 and 6 are substantially compressed, but this will affect the density of the aerogel only if the strain is irreversible. Studies of mechanical compression of aerogels [23] have shown that if the density is raised beyond a threshold value, qy , the strain is largely irreversible. For an initial density of q0 , the threshold is given approximately by [24] qy q0 expð1=mÞ:
ð62Þ
Given q ¼ 100 kg/m3 and m ¼ 3:5, we find qy 133 kg/m3 . The horizontal arrow in Fig. 15 represents this threshold, and indicates that the density in disks 5 and 6 has exceeded it. This explains the observed velocity distribution in Fig. 6: disks 1–4 have not passed the threshold, so their final densities are equal at about 100 kg/m3 , but disks 5 and 6 have been irreversibly compressed
during RSCE, so their final densities are raised just about qy . A comparable calculation indicates that there would have been no irreversible strain if the initial modulus of the gel had been 2.1 MPa (i.e., equal to that of the final aerogel). By performing the simulation with various assumed values for H, we find that the highest modulus the gel could have that would allow disk 5 to reach qy is H ¼ 0:5 MPa. That is, the heat treatment could not have raised the modulus above that value in the temperature range where maximum compression occurs (T < 150 °C), or disk 5 would have retained a density of 100 kg/m3 . The fact that part of the aerogel is densified indicates that the cycle used in the experiment is not optimal. In the Section 5, an optimal heating rate is calculated for this type of gel.
5. Optimal cycle Here we assume that the pressure in the autoclave is 13.8 MPa, as in experiment RSCE B-271, but we calculate a heating rate that avoids irreversible compression of any part of the gel. The goal is to prevent the strain in the gel from exceeding the threshold of irreversible compression. According to Eq. (62), the limiting strain is qy 1 ezy ¼ ln ð63Þ
: m q0 When m ¼ 3:5, this strain is ezy ¼ 0:29 and qy ¼ 133 kg/m3 . To prevent ez from exceeding ezy , Eq. (56) indicates that the maximum heating rate is dT 1 DH 6 ¼ : dt maF s maF gF L2
Fig. 15. Calculated density at locations corresponding to disks in Fig. 2, when YoungÕs modulus ¼ 0:1 MPa.
ð64Þ
Using Eqs. (52) and (60), with m ¼ 3:5, D ¼ 35 nm2 , L ¼ 2:33 cm, Eq. (64) has been evaluated for several values of H. Fig. 16 shows that the rate used in experiment RSCE B-271 was much too fast, if the modulus of the gel was E ¼ 0:1 MPa (H ¼ 0:11 MPa); on the other hand, if the modulus were initially equal to that of the final aerogel (E ¼ 2:1 MPa), the cycle could have been much faster. Since the density of disk 5 barely exceeded the threshold of irreversible deformation, it is reasonable to infer that the modulus of the gel
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272
Fig. 16. Comparison of thermal cycle used in experiment RSCE B-271 with optimal cycles calculated assuming E ¼ 0:1 or 2.1 MPa.
Fig. 17. Comparison of thermal cycle used in experiment RSCE B-271 with optimal cycle calculated assuming H ¼ 0:5 MPa.
reached about H ¼ 0:5 MPa in the temperature range of maximum strain. As shown in Fig. 17, the optimal cycle is actually shorter than the one used in the experiment, but the rate is slower at low temperatures. More precise predictions would require knowledge of the evolution of H as a function of temperature, and the dependence of permeability on pressure.
6. Summary The major cause of heterogeneity in RSCE aerogels is deformation caused by venting; syneresis can create a denser region near the exterior surface, but it is a relatively minor effect. Measurements of the venting rate for molds containing
271
either pure methanol or a silica gel indicate that the rate of escape is approximately equal to the rate of thermal expansion of the fluid. Using this as a boundary condition, a one-dimensional analysis has been obtained that describes the distribution of pressure and strain within the aerogel in an RSCE mold. The strain is predicted to arise mainly at high temperatures, where the expansion of the fluid increases rapidly. Therefore, the strain of the aerogel can be reduced by decreasing the heating rate in inverse proportion to the product of the viscosity and thermal expansion coefficient of the fluid. A sample calculation shows that this allows reasonable heating rates to be used without causing permanent strain. However, this calculation assumes that the fluid is venting uniformly over the surface of the mold, whereas the original molds vent along a line. Use of a fritted mold has been shown to yield the expected benefit: more uniform strain of the gel, resulting in more uniform final density. Nevertheless, experiment RSCE B-271 produced gels with some densification in the region near the frit. The present analysis shows that the measured densities can be explained by taking account of the low modulus of the initial gel. Using the measured modulus, the model predicts that the gel will be compressed beyond the limit of irreversible strain in precisely that region of the sample where densification was observed. The model permits calculation of an optimal heating cycle (i.e., the highest rate that avoids densification). Using a reasonable estimate of the modulus of the gel at elevated temperature, the optimal cycle is found to be shorter than the one used in RSCE B-271, but uses a slower heating rate at low temperatures. Unfortunately, the project that motivated this study has been cancelled, so we cannot test the predicted optimal cycle using close-loop control of the system. Nevertheless, the experiments to date are in good agreement with the theory, and we expect that highly uniform aerogels could be made by the RSCE process.
Acknowledgements The authors are indebted to Hang-Shing Ma for his help with the beam-bending measurements.
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This work was performed under the auspices of the US Department of Energy by University of California, Lawrence Livermore National Laboratory under contract no. W-7405-ENG-48. That work was described in ÔOptimization of the Rapid Supercritical Extraction Process for AerogelsÕ, UCRL-JC-145444.
References [1] J.F. Poco, P.R. Coronado, R.W. Pekala, L.W. Hrubesh, in: R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton, S.I. Zones (Eds.), Microporous and Mesoporous Materials, MRS Symposium Proceedings, vol. 431, Materials Research Society, Pittsburgh, PA, 1996, p. 297. [2] G.W. Scherer, H. Hdach, J. Phalippou, J. Non-Cryst. Solids 130 (1991) 157. [3] G.W. Scherer, J. Non-Cryst. Solids 145 (1992) 33. [4] G.W. Scherer, J. Sol–Gel Sci. Tech. 3 (1994) 127. [5] T. Woignier, G.W. Scherer, A. Alaoui, J. Sol–Gel Sci. Tech. 3 (1994) 141. [6] J. Gross, P.R. Coronado, L.W. Hrubesh, J. Non-Cryst. Solids 225 (1998) 282. [7] G.W. Scherer, J. Non-Cryst. Solids 142 (1-2) (1992) 18. [8] G.W. Scherer, J. Sol–Gel Sci. Tech. 1 (1994) 169. [9] G.W. Scherer, J. Non-Cryst. Solids 201 (1996) 1. [10] L.W. Hrubesh, C.T. Alviso, in: C.J. Brinker, D.E. Clark, D.R. Ulrich (Eds.), Better Ceramics Through Chemistry III, Materials Research Society Symposium Proceedings,
[11] [12] [13] [14]
[15] [16]
[17] [18] [19] [20] [21]
[22] [23] [24]
vol. 121, Materials Research Society, Pittsburgh, PA, 1988, p. 703. S.-Y. Wang, N.-L. Wu, J. Non-Cryst. Solids 224 (1998) 259. H.N. Holmes, W.E. Kaufmann, H.O. Nicholas, J. Am. Chem. Soc. 41 (1919) 1329. G.W. Scherer, J. Non-Cryst. Solids 108 (1989) 18. G.W. Scherer, C. Alviso, R. Pekala, J. Gross, in: R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton, S.I. Zones (Eds.), Microporous and Mesoporous Materials, MRS Symposium Proceedings, vol. 431, Materials Research Society, Pittsburgh, PA, 1996, p. 497. G.W. Scherer, J. Non-Cryst. Solids 215 (2–3) (1997) 155. R.W. Pekala, L.W. Hrubesh, T.M. Tillotson, C.T. Alviso, J.F. Poco, J.D. LeMay, in: K. Sieradzki, D.J. Green, L.J. Gibson (Eds.), Mechanical Properties of Porous and Cellular Materials, Materials Research Society, Pittsburgh, PA, 1991, p. 197. J. Gross, G. Reichenauer, J. Fricke, J. Phys. D 21 (1988) 1447. G.W. Scherer, J. Non-Cryst. Solids 109 (1989) 171. C.J. Brinker, G.W. Scherer, Sol–Gel Science, Academic Press, New York, 1990 (Chapter 7). J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, 1986. P.E. Liley, T. Makita, Y. Tanaka, in: C.Y. Ho (Ed.), Properties of Inorganic and Organic Fluids, Hemisphere, New York, 1988 (Chapter 20). K.C. Chao, R.A. Greenkorn, Thermodynamics of Fluids, Marcel Dekker, New York, 1975, p. 302. G.W. Scherer, D.M. Smith, X. Qiu, J.M. Anderson, J. Non-Cryst. Solids 186 (1995) 316. D.M. Smith, G.W. Scherer, J.M. Anderson, J. Non-Cryst. Solids 188 (1995) 191.
Optimization of the rapid supercritical extraction process for aerogels George W. Scherer
a,*
, Joachim Gross a,1, Lawrence W. Hrubesh b, Paul R. Coronado b
a
b
Princeton University, CEOR/PMI, Eng. Quad. E-319, Princeton, NJ 08544, USA Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 USA Received 17 October 2001; received in revised form 28 February 2002
Abstract A partial differential equation is derived that describes the pressure developed in the pores of a gel during the rapid supercritical extraction process. A comparative analysis of the strains caused by syneresis and expansion of the fluid, respectively, suggests that the latter is the dominant effect for this process. Experimental results indicate that the rate of leakage from the mold is equal to the rate of volumetric expansion of the fluid, so this was used as the boundary condition for the calculation. An analytical solution is obtained for the strain produced in a purely elastic gel. The strain is found to develop most rapidly at high temperatures, where the thermal expansion of the fluid increases sharply. The model predicts a temperature dependent heating rate that can be used to avoid irreversible strains by compensating for the increase in thermal expansion coefficient. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The rapid supercritical extraction process (RSCE) permits extraordinarily rapid preparation of aerogels [1]. The wet gel is placed into a mold, which is then inserted into an autoclave. As the temperature of the autoclave rises, the temperature and pressure in the mold increase until the fluid begins to leak from the mold; deformation of the gel is largely prevented because it is constrained *
Corresponding author. Tel.: +1-609 258 5680; fax: +1-609 258 1563. E-mail address: [email protected] (G.W. Scherer). 1 Present address: Zentrum Fuer Medizinische Forschung, Waldhoernlestr. 22, Tuebingen 72072, Germany.
inside the mold, so the temperature can be raised beyond the critical point much faster than in conventional supercritical drying. Unfortunately, the resulting aerogels often exhibit spatially varying properties. The inhomogeneity of the aerogels is believed to result from pressure gradients created as the vapor is vented from the mold. Therefore, a mold was developed with a fritted plate on one end, so that the pressure would be released uniformly over a larger area. The purpose of this paper is to analyze the gradients in that type of mold, and to identify the parameters that must be controlled to minimize them. The analysis is an extension of earlier studies of stresses developed during thermal expansion of the pore liquid [2,3] and depressurization of the autoclave [4,5].
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 3 7 9 - 0
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2. Experimental procedure 2.1. Rapid supercritical extraction process The original RSCE mold was cylindrical with a diameter of 29.3 mm and length of 23.3 mm. The mold was filled with a sol (described in Section 2.3) and the end plates were bolted on. The mold was placed inside an autoclave that was filled with methanol; the pressure in the liquid was pumped to 2000 psi (13.8 MPa) then the assembly was heated. As the temperature rose, the pressure in the mold eventually forced the lid open slightly, so that fluid leaked out. Two experiments were performed to study the rate of leaking by capturing and weighing the effluent. In one case (experiment Para-A-35), the mold contained only methanol,
and in another (Para-A-41) it contained a sol of the type described below. A new cylindrical mold was made with the same dimensions and design, except that a steel frit was placed over one end. The samples were subjected to the temperature–pressure history shown in Fig. 1, where the pressure immediately ramped to about 2000 psi (13.8 MPa) before the temperature was raised. This experiment is identified as RSCE B-271. In both experiments, the accuracy is about 3% for the pressure and 2% for the temperature measurements. 2.2. Density profiles To ascertain the uniformity of the aerogels, the acoustic pulse velocity was measured on disks cut from the cylindrical gels. The disks were 30 mm in diameter and about 2 mm thick, as shown in Fig. 2. The acoustic velocity was measured with 180 kHz transducers, as explained in detail in Ref. [6]. The accuracy of the sound velocities is about 2%, or about 2–10 m/s. 2.3. Permeability and modulus
Fig. 1. Temperature and pressure in the fritted mold during RSCE experiment.
Rods of silica gel were made using the same recipe as for RSCE. Solution A was prepared by mixing 35.98 g of methanol (99.8þ% from Aldrich) with 30.06 g of tetramethoxysilane (TMOS,
Fig. 2. Schematic (a) and photo (b) of fritted mold for RSCE.
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99þ% from Aldrich). Solution C was made using 196.62 g methanol and 3.41 g NH4 OH, and solution B was prepared using 21.84 g H2 O, 36.14 g methanol and 1.02 g of solution C. Solution B was then poured into solution A and the stirring was continued for 10 more minutes. The homogeneous gel solution was poured into polystyrene molds of 7.9 mm internal diameter and the molds were capped with rubber septa, then the gels were cured in the incubator at 49 °C. One or two molds were taken out of the incubator after aging for 1, 2, 3, 4, 6, 8, and 12 days, respectively. The gel beams were transferred from the mold to the 20 ml test-tubes and were washed with methanol five times within 3 days. The gels were tested by beam-bending [7–9] using a 50 g load cell with methanol as the bath solution. The span for three-point bending was 82.2 mm and the deflection was set at 1.5 mm, which corresponds to 6 1% strain in the gel beams. Upon application of the deflection, a pressure gradient is created in the methanol in the pores, so the liquid begins to flow within the beam and to exchange with the surrounding bath. As the pore pressure equilibrates, the force required to sustain a constant deflection decreases; by fitting the force decay to the theoretical expression [8], the permeability, elastic modulus, and PoissonÕs ratio for the gel can be obtained.
3. Results 3.1. Beam-bending The gels are relatively compliant, so some initial loading data (for about 1 s) failed to be recorded
261
due to vibration in the beam-bender and the small loading value (around 0.3 g force). The summary of the beam-bending results is shown in Table 1. The permeability is calculated from the hydrodynamic relaxation time by taking 0.544 mPa s as the viscosity of methanol. A typical plot of the hydrodynamic relaxation curve measured in the beam-bending experiment is shown in Fig. 3. The fit is reasonably good, but the load is small because of the compliance of the gel, and this is responsible for the scatter in the results reported in Table 1. The permeability of the gel is 32 10 nm2 and the YoungÕs modulus is 0.1 MPa; neither property shows systematic variation with age, although the modulus may be rising at 12 days of aging. As the gel is heated in the autoclave, the modulus will increase; measurements of aerogels made from this recipe indicate that E ¼ 2:1 MPa. However, since the volume of the gel does not change appreciably, the permeability (which depends on the porosity and pore size) is expected to remain nearly constant. 3.2. Density gradients in wet gels Before considering the shrinkage during RSCE, we will discuss the density gradients observed in gels prepared under ambient conditions. The refractive index was measured on wet gels using methods described elsewhere [10]. Wet gels show a region within a few millimeters of the outer surface where the density is higher than in the interior, even if the sols, gels and molds are maintained at room temperature. It was recently reported that the shrinkage of a gel is related to surface charges and solvent polarity [11], so shrinkage near the surface of the gel might result from contamination
Table 1 Summary of beam-bending data Aging (days)
Beam-bending Poisson ratio
Hydrodynamic s (s)
Shear modulus G (MPa)
Elastic modulus E (MPa)
Permeability D (nm2 )
Gel diameter d (mm)
1 2 3 4 6 8 12
0.169 0.142 0.105 0.274 0.127 0.147 0.138
5890 3770 6570 2940 3900 4580 1970
0.0436 0.0366 0.0357 0.0412 0.0447 0.0498 0.0695
0.102 0.0835 0.0789 0.105 0.101 0.114 0.158
21.6 43.6 27.6 30.9 35.3 25.0 41.9
7.86 7.81 7.76 7.79 7.79 7.72 7.62
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where gF is the viscosity of the pore fluid, D is the permeability of the network, HG is the longitudinal viscosity of the drained network, and b ¼ ð1 þ mÞ= ½3ð1 mÞ, where m is PoissonÕs ratio; In is a modified Bessel function of the first kind of order n. For a rigid gel (large HG ) or a highly permeable gel (large D), a is small and Eq. (1) reduces to lim e_ðrÞ ¼ 3e_s a!0
Fig. 3. Relaxation of load, W, during beam-bending experiment performed on gel aged 8 days; wVE represents viscoelastic part of relaxation.
of the pore liquid by corrosion of the mold; that is, ions released by corrosion of the mold could alter the surface charges on the gel network and thereby influence dilatation. However, chemical analysis of thin gels (which have maximal contact with vessel walls) by ICP has revealed only tens of ppm of contamination, which is not likely to cause appreciable increase in shrinkage rate. It is known that the rate of syneresis is affected by the size of the gel, because shrinkage requires escape of the liquid from the network [12,13]. Within a large gel, the syneresis rate is greater near the exterior surface, where the pore liquid escapes freely from the network. The pressure distribution in the pore liquid has been calculated for a cylindrical gel with a viscous or viscoelastic network [13]. For the viscous case, the volumetric strain rate in a cylindrical gel with radius r0 is abI0 ðar=r0 Þ e_ðrÞ ¼ 3e_s ; ð1Þ aI0 ðaÞ 2ð1 bÞI1 ðaÞ where e_s is the linear syneresis strain rate inherent to the network (which is equal to the observed rate only in very thin gels, where flow of the liquid does not inhibit contraction). The parameter a is defined by sffiffiffiffiffiffiffiffiffiffi gF r02 a¼ ; ð2Þ DHG
ð3Þ
so the volumetric strain rate approaches the inherent rate (the factor of 3 appears because e_s is a linear strain rate) when a is small. For a wide variety of silica and organic gels examined in our lab [14,15], we find that 0:15 6 m 6 0:25; if m ¼ 0:2, then b ¼ 0:5, and Eq. (1) becomes aI0 ðar=r0 Þ e_ðrÞ 1 ¼ : ð4Þ 2 aI0 ðaÞ I1 ðaÞ 3e_s This result is plotted in Fig. 4 for three values of a. When a is small, the contraction rate is uniform throughout the cylinder, but as a increases, the contraction is more severely inhibited deep within the body; therefore, the density of the exterior will be increased by syneresis compared to the interior. Based on the beam-bending data for the sample shown in Fig. 3, the value of a for a cylindrical gel with a bulk density of qb ¼ 0:1 g/cm3 and r0 ¼ 2 cm is expected to be on the order of 0.1, so the exterior is not expected to have a significantly higher shrinkage rate than the interior.
Fig. 4. Volumetric strain rate normalized by inherent syneresis rate versus radial position for several values of a. Shrinkage rate is greatest at exterior surface.
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The only temperature-dependent factor on the right side of Eq. (4) is a, which depends on the ratio gF =HG ; it seems likely that the activation energy for viscous flow of the network will be somewhat higher than for the pore liquid, in which case a will increase with temperature, and the density gradients will be more severe. However, even if our estimate of a is low, Fig. 4 indicates that the variation in shrinkage rate varies roughly parabolically with radius, whereas the observed density increase was pronounced only within a few millimeters of the surface. Therefore, it appears unlikely that syneresis makes a significant contribution to the overall density gradients seen in aerogels made by RSCE. 3.3. Density gradients in RSCE gels The spatial variation in acoustic velocity in an RSCE aerogel prepared in a conventional (unfritted) mold is illustrated in Fig. 5, which is taken from Ref. [6]. The gradients are most pronounced near the upper perimeter of the gel, near the place where the vapor leaks out of the mold. Away from the end, the velocity is almost constant, so the density is probably also nearly constant. There are strong radial gradients in temperature in the mold, but neither those gradients nor the effect of syneresis could account for the shape of the variation seen in Fig. 5. It must be the venting process itself that causes the greatest variation in density.
Fig. 5. Longitudinal sound velocity as function of position measured in an RSCE aerogel made in a conventional (unfritted) mold.
263
The longitudinal sound velocity, V, measured in an aerogel made in the fritted mold is shown in Fig. 6; the disk numbers correspond to the slices shown in Fig. 2. The velocity is related to the longitudinal modulus by H ¼ qb V 2 ;
ð5Þ
where H ¼ ð1 mÞE=½ð1 2mÞð1 þ mÞ, E is YoungÕs modulus of the gel network, m is PoissonÕs ratio, and qb is the bulk density. Extensive measurements of the elastic modulus of aerogels [16] indicate that the modulus varies with density according to a power law of the form m
H ¼ H0 ðqb =q0 Þ ;
ð6Þ
where H0 is the modulus when the density of the network is q0 , and the exponent typically lies in the range 2:5 6 m 6 4. From Eqs. (5) and (6), we find that the velocity is related to the density of the aerogel by qb ¼ ðqm0 =H0 Þ
1=ðm1Þ
V 2=ðm1Þ :
ð7Þ
Since m is typically near 3.5 for silica aerogels [16], V is nearly linearly proportional to the density of the gel. Fig. 6 shows that the density is much more uniform across the radius of the disk than for gels made from RSCE molds without a fritted end (Fig. 5). Nevertheless, disks 5 and 6 (near the frit) are significantly denser than the rest of the aerogel. PoissonÕs ratio for silica gels and aerogels is typically found to be m 0:2 [17], so H 1:1E;
Fig. 6. Longitudinal sound velocity measured in aerogel made in fritted mold shown in Fig. 2, subjected to the thermal cycle given in Fig. 1.
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since E0 2:1 MPa for this type of aerogel when q0 ¼ 100 kg/m3 , we estimate that H0 ¼ 2:3 MPa. Eq. (7) can be used to estimate the densities of the slices in the disks in Fig. 2: if m ¼ 3:5, then the density in disk 6, where V 230 m/s, is 140 kg/m3 and the density in disk 2, where V 150 m/s, is 99 kg/m3 . Thus, disks 1–4 have about the expected density for this recipe, while disks 5 and 6 have densities increased by 30–40%.
4. Analysis of RSCE 4.1. Elastic case The data in Figs. 5 and 6 indicate that the density gradients in the aerogel result primarily from compression of the network during venting, so we will begin by calculating the strain caused by that phenomenon alone. We will consider only the uniaxial flow that is expected to occur in a fritted mold, shown schematically in Fig. 7. The mold vents uniformly across the fritted surface at z ¼ L when the fluid pressure, PF ðz; tÞ, satisfies
The strains are ex ¼ ey ¼ ef þ
1 ½ð1 mÞrx mrz E
ð10Þ
and 1 ½rz 2mrx ; ð11Þ E where E and m are YoungÕs modulus and PoissonÕs ratio for the gel network. The free strain ef is the linear strain that would occur in the absence of stress; in a thermal expansion problem it is the product of the thermal expansion coefficient and the change in temperature. Let the expansion of the mold caused by thermal expansion and internal pressure be e0 , where ez ¼ ef þ
e0 ¼
hPF i þ aM DT ; RM
ð12Þ
ð8Þ
where RM is the rigidity of the mold (which depends on the modulus and dimensions of the mold), aM is the linear thermal expansion coefficient of the mold, and hPF i is the average pressure in the fluid inside the mold. Then ex ¼ ey ¼ e0 and rz ð13Þ ez ¼ 3bef þ ð1 3bÞe0 þ ; H
where P0 is the threshold pressure at which the mold begins to leak. To find the stresses in a gel, the first step is to solve the corresponding thermal stress problem [18]. In the present case, the flow is only along z, so there is symmetry in the x and y directions. The stresses are
where H is the longitudinal modulus of the gel network and b was defined previously. The axial strain is Z 1 L ez dz ¼ e0 : ð14Þ L 0
rx ¼ ry :
Substituting Eq. (13) into Eq. (14) leads to
PF ð1; tÞ P P0 ;
ð9Þ
rz ¼ 3Kðe0 hef iÞ
ð15Þ
and ez ¼ e0 þ 3bðef hef iÞ;
ð16Þ
where K ¼ bH is the bulk elastic modulus of the network and Z 1 L hef i ¼ ef dz: ð17Þ L 0 The volumetric strain is e ex þ ey þ ez ¼ 3e0 þ 3bðef hef iÞ:
ð18Þ
For a gel, the free strain rate is given by [2,18] Fig. 7. Cross-section of mold having length L and radius r0 ; dimensionless length is u ¼ z=L.
ef ¼ es þ aS DT þ
PF 3K
ð19Þ
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which includes contributions from syneresis (es ), thermal expansion of the gel network (aS ¼ thermal expansion of the solid), and pressure in the pore fluid. From Eqs. (16) and (19), the axial strain is ez ¼ e0 þ
PF hPF i H
ð20Þ
so the volumetric strain is e ¼ 3e0 þ
PF hPF i : H
ð21Þ
4.2. Continuity equation
ð22Þ
where qF is the density of the fluid, q is the volume fraction of solids in the network, and VF is the volume of fluid per gram of network; J is the flux of fluid, which is related to the pressure in the fluid by DarcyÕs law [19,20]: J ¼
D rPF ; gF
ð23Þ
where D is the permeability of the network and gF is the viscosity of the fluid. It has been shown that [2] e_ ¼ ð1 qÞ
V_F þ 3aS qT_ VF
ð24Þ
so Eq. (22) becomes q_ F þ
qF 1 r ðqF J Þ: ðe_ 3aS qT_ Þ ¼ 1q 1q
ð25Þ
To link Eqs. (21) and (25), we need the relation between qF and PF (i.e., the equation of state of the fluid). For the sake of simplicity we adopt the following linear equation (which is adequate for the liquid state), q_ F P_ F ¼ 3aF T_ ; qF KF
The general solution for the uniaxial case is obtained by solving Eqs. (21), (25) and (26), but the problem is highly non-linear, and can only be treated numerically. A useful approximation is obtained if we consider only the case where the quantity qF D=gF is weakly dependent on position, so that it can be removed from the divergence operator on the right side of Eq. (25). Then the continuity equation takes the form 1q 1 1 3 D o2 PF þ e_T ¼ hP_ F i ; P_ F KF H H RM gF oz2 ð27Þ where
The equation of continuity (or conservation of mass) is given by [4] 1 o 1 ðq VF Þ ¼ r ðqF J Þ; VF ot F 1q
265
ð26Þ
where KF is the bulk modulus (the reciprocal of the compressibility) of the fluid.
e_T ¼ 3½qaS þ ð1 qÞaF aM T_ :
ð28Þ
Finally, we introduce the hydrodynamic relaxation time, s, defined by gF L2 DH and the dimensionless time Z t dt h¼ 0 s
s¼
ð29Þ
ð30Þ
so that dt ¼ s dh. With these variables, Eq. (27) can be written as oPF ð1 qÞH ohPF i 3H oeT 1 1þ H KF oh RM oh oh o2 PF ¼ : ð31Þ ou2 Since the modulus of the gel is completely negligible compared to that of the mold, H RM , the quantity H =RM will be neglected. The same argument could be made for the quantity H =KF , but we will retain that term for the moment.
4.3. Boundary condition There is some threshold pressure, P0 , beyond which the mold begins to leak. To get an idea of the dependence of the leakage rate on pressure, we have examined data for the rate of release of pure methanol from an RSCE mold (experiment ParaA-35). The internal and external temperature and pressure were measured and the escaping methanol
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was collected, with the results shown in Fig. 8. Segment H of the pressure curve shows the rapid increase in pressure as the methanol (which initially fills the mold completely) is heated. The pressure in the liquid in that stage is given by
nearly incompressible, the amount of fluid that must escape is the difference in volume between the contents of the mold and the mold itself,
PF ¼ 3KF ðaF aM ÞDT ;
where VM is the volume of the mold. Therefore, the mass of methanol collected is expected to be
ð32Þ
where aF and aM are the thermal expansion coefficients of the fluid and the metal mold, respectively. For methanol, aF 4:3 104 =°C and KF ¼ 7:7 108 Pa; for 304 stainless steel, aM ¼ 1:7 105 =°C, so Eq. (32) predicts dPF =dT
0:95 MPa=°C ¼ 138 psi/°C, whereas the measured slope is 123 psi=°C ¼ 0:85 MPa=°C. This is reasonable agreement in view of the existence of temperature gradients and, more importantly, the probable decrease in the bulk modulus of liquid methanol at high pressure. Methanol begins to be collected at an appreciable rate after the pressure reaches 2150 psi, and the rate of collection increases linearly with temperature while the pressure remains nearly constant; near the critical point (240 °C), the pressure and leakage rate jump. The mass of fluid escaping is dMF dV ; ¼ qF dt dt
ð33Þ
where V is the volume of fluid escaping. As the temperature increases, the volumetric expansion of the contents of the mold is 3aF DT ; since the fluid is
dV dT ¼ VM ð3aF 3aM Þ ; dt dt
ð34Þ
dMF dT ; ¼ qF VM ð3aF 3aM Þ dt dt
ð35aÞ
dMF dT ;
3qF VM aF dt dt
ð35bÞ
where the approximation is justified by the small expansion of steel compared to the fluid. The thermal expansion coefficient is defined by aF ¼
1 dqF 3qF dT
ð36Þ
so Eq. (35b) reduces to dMF dq dT
VM F dt dT dt
ð37Þ
or dMF dq
VM F : dT dT
ð38Þ
For methanol [3], the density along the saturation line is given approximately by 0:360 513 T ðKÞ qF ðg=cm3 Þ ¼ 0:2722 þ ð39Þ 1359 so dqF 0:027
: dT ð513 T ðKÞÞ0:64
Fig. 8. Pressure in mold (reactor) and weight of methanol condensed at exit. Arrows show sequence of cycle; portion labeled H represents pressure increase as liquid methanol is heated before any liquid escapes from mold.
ð40Þ
The measured rate of mass escaping is obtained by smoothing, then differentiating the curve in Fig. 8. Using those data for dMF =dT along with Eq. (40), Eq. (38) predicts that the quantity shown in Fig. 9 should be equal to VM . We see that Eq. (38) is obeyed from ambient temperatures to about 200 °C; the constant value of 12 cm3 agrees closely with the volume of the mold (2.5 cm dia 2:5 cm thick ¼ 12:3 cm3 ). This result indicates that the fluid in the mold is essentially incompressible, so that the rate of escape from the mold equals the rate of expansion, once the pressure is
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272
Fig. 9. Test of Eq. (38) using data from same experiment as Fig. 8.
sufficient to force open a leak. Near the critical point, the fluid expands more rapidly than it can escape (causing the downward curvature in Fig. 9), so the pressure rises. A similar experiment was performed (experiment Para-A-41) where the mold contained a sol chosen to yield a gel with a density of 0.1 g/cm3 . As shown in Fig. 10, the pressure in the mold jumps by 75% when gelation occurs, but there is no corresponding change in the rate of collection of fluid. The increase in pressure is a result of the increase in viscosity upon gelation. Since the sol is mostly methanol, Eq. (38) should still be obeyed in this case, and Fig. 11 shows that it is; the constant value is again 12 cm3 . There is a jump in Fig. 11 at 75 °C, when gelation occurs, but thereafter the plot remains flat almost all the way to the critical temperature. These results indicate that the appropriate boundary condition for calculating the stress in the gel during RSCE is to set the flux out of the mold, J, equal to the volumetric expansion of the gel: J jz¼L ¼
1 dV ¼ A dt
3VM dT ½ð1 qÞaF þ qaS aM ; A dt ð41Þ
where A is the area of the leak, and the quantity in brackets includes the expansion of the fluid, the
267
Fig. 10. Pressure in mold (reactor) and weight of fluid condensed at exit. Arrows show sequence of cycle. Label L indicates the onset of leakage and label G indicates the point at which gelation occurred.
Fig. 11. Test of Eq. (38) using data from same experiment as Fig. 10, where mold contains a gelling solution.
solid (i.e., the gel network), and the mold. Using Eq. (23), the boundary condition becomes D dPF 3VM dT : ¼ ½ð1 qÞaF þ qaS aM g dz dt A F
z¼L
ð42Þ 4.4. Elastic solution We seek the solution of Eq. (31) subject to the boundary conditions given by Eq. (42) and D dPF ¼0 ð43Þ g dz F
z¼0
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which indicates that there is no flux across the bottom end of the mold. If we choose to set time to zero at the moment when the leak begins, the initial condition is PF ðz; 0Þ ¼ P0 :
ð44Þ
The solution is PF ðu; hÞ ¼ P0 þ H
Z
h
Xðu; h h0 Þ 0
oeT 0 dh ; oh0
ð45Þ
where the thermal strain is eT ¼ 3½ð1 qÞaF þ qaS aM ðT TR Þ;
ð46Þ
where TR is room temperature. The relaxation function that appears in Eq. (45) is Xðu; hÞ 2
1 X
n
ð1Þ cosðnpuÞ expðn2 p2 hÞ:
ð51Þ
For methanol, we can use Eq. (39) for qF , and find aF from Eq. (36). Data are available [21] for the viscosity of methanol in the range of temperature and pressure used in this experiment, and they are well represented by 3:92 44:6 gF ðPa sÞ ¼ : ð52Þ T ðKÞ The resulting values for gF aF are shown in Fig. 12. Since this quantity rises rapidly as the critical point approaches, Eqs. (49) and (50) indicate that the strain in the gel will rise rapidly in that region. Suppose that instead of using a constant heating rate, we let the rate vary as
ð47Þ
oT c ¼ ; ot aF gF
ð48Þ
where c is a constant. In this case, Eq. (50) indicates that oeT =oh is roughly constant, so Eq. (49) becomes 1 2 oeT X ð1Þn cosðnpuÞ ez 2 ½1 expðn2 p2 hÞ: p oh n¼1 n2
n¼1
The average pressure in the mold is Z 1 hPF i ¼ PF ðu; hÞ du ¼ P0
oeT / gF aF : oh
0
so the average pressure remains constant, in keeping with Fig. 10. The axial strain in the gel network is found from Eqs. (20) and (45); neglecting the thermal expansion of the mold (e0 ), the result is Z h oeT ez ¼ Xðu; h h0 Þ 0 dh0 : ð49Þ oh 0
ð53Þ
ð54Þ
If the duration of the experiment is long compared to the hydrodynamic relaxation time of the gel, we can assume that t s, so h 1, and Eq. (54) reduces to
4.5. Effect of heating rate on deformation The principle problem with RSCE is the deformation of the final aerogel, so we are most interested in the implications of Eq. (49). The strain in the gel is related to the quantity oeT oeT gF L2 oT ¼s ¼ ½3ð1 qÞaF þ 3qaS 3aM oh ot DH ot 2 gF L oT ð3aF Þ ; ð50Þ
DH ot where the approximation is justified by the relatively small values of aS and aM . If the heating rate is constant, then
Fig. 12. Product of viscosity and thermal expansion coefficient for methanol along the saturated liquid/vapor line.
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272 1 2 oeT X ð1Þn cosðnpuÞ p2 oh n¼1 n2 1 oeT 2 1 u
: 2 oh 3
269
ez ðu; hÞ
ð55Þ
The maximum strain occurs at the end where the leak is occurring (u ¼ 1): ez ð1; hÞ
1 oeT oT
aF s ; 3 oh ot
h 1;
ð56Þ
where the second equality follows from Eq. (50). This indicates that the strain in the network can be kept to a safe level by choosing the heating rate such that a safe constant value of oeT =oh is preserved. This analysis indicates that the deformation of the aerogel is minimized by those factors that decrease s. The properties of the gel that are most important are the permeability (D) and elastic modulus (H), both of which increase with aging and thereby reduce s. If the thermal cycle included a hold above the gel point and before the onset of leaking, it would serve to coarsen the gel, raising D and H. 4.6. Simulation of the experiment We want to evaluate Eq. (49) for the thermal cycle shown in Fig. 1, which requires that we know the viscosity and thermal expansion coefficient of the fluid, and the modulus and permeability of the gel. The viscosity is given by Eq. (52) and the density of the fluid can be calculated from data for the compressibility factor, f, defined by f¼
PF Vm ; Rg T
Fig. 13. Compressibility factor for methanol at 141 bars (2080 psi, 14.3 MPa).
qF ¼
M w M w PF ; ¼ Vm fRg T
ð59Þ
where Mw is the molecular weight (for methanol, Mw ¼ 32). The density calculated in this way is accurately fit by aF ¼ 2:047 103 5:0142 105 T þ 9:0238 107 T 2 6:5805 109 T 3 þ 1:6248 1011 T 4 :
ð60Þ
The properties obtained from Eqs. (52) and (60) are shown in Fig. 14. The permeability of the gel can be assumed to remain constant near the measured value of 32 nm2 , but the elastic modulus changes from 0.1 MPa in the initial wet gel to 2.1 MPa in the aerogel, and the evolution of that property in the autoclave is not known. For the following
ð57Þ
where Vm is the molar volume and Rg is the gas constant; f ¼ 1 for an ideal gas. Data are available [22] for f in the range of temperature and pressure used in the RSCE experiment, and a good fit is provided by f ¼ 0:5521 1:6878 104 T ð°CÞ þ 0:3107 tanhð0:0203T ð°CÞ 6:391Þ;
ð58Þ
where T is in °C (in this equation only); the curve is shown in Fig. 13. The density of the fluid is
Fig. 14. Estimated thermal expansion coefficient and viscosity for the fluid in the aerogel during the thermal history shown in Fig. 1.
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simulations, we will assume that the modulus is constant and equal either to 0.1 or 2.1 MPa, in order to set bounds on the behavior. The axial strain, ez , is obtained by numerical solution of Eq. (49), using the temperature and pressure history from Fig. 1, along with the physical properties determined above. Since the strains in the lateral directions are constrained by the mold, the axial and volumetric strains are equal, and the density of the gel is related to the axial strain by q ¼ q0 expðez Þ:
ð61Þ
Fig. 15 shows the density as a function of temperature at the positions corresponding to the disks in Fig. 2, when E ¼ 0:1 MPa. The calculations in Fig. 15 indicate that disks 5 and 6 are substantially compressed, but this will affect the density of the aerogel only if the strain is irreversible. Studies of mechanical compression of aerogels [23] have shown that if the density is raised beyond a threshold value, qy , the strain is largely irreversible. For an initial density of q0 , the threshold is given approximately by [24] qy q0 expð1=mÞ:
ð62Þ
Given q ¼ 100 kg/m3 and m ¼ 3:5, we find qy 133 kg/m3 . The horizontal arrow in Fig. 15 represents this threshold, and indicates that the density in disks 5 and 6 has exceeded it. This explains the observed velocity distribution in Fig. 6: disks 1–4 have not passed the threshold, so their final densities are equal at about 100 kg/m3 , but disks 5 and 6 have been irreversibly compressed
during RSCE, so their final densities are raised just about qy . A comparable calculation indicates that there would have been no irreversible strain if the initial modulus of the gel had been 2.1 MPa (i.e., equal to that of the final aerogel). By performing the simulation with various assumed values for H, we find that the highest modulus the gel could have that would allow disk 5 to reach qy is H ¼ 0:5 MPa. That is, the heat treatment could not have raised the modulus above that value in the temperature range where maximum compression occurs (T < 150 °C), or disk 5 would have retained a density of 100 kg/m3 . The fact that part of the aerogel is densified indicates that the cycle used in the experiment is not optimal. In the Section 5, an optimal heating rate is calculated for this type of gel.
5. Optimal cycle Here we assume that the pressure in the autoclave is 13.8 MPa, as in experiment RSCE B-271, but we calculate a heating rate that avoids irreversible compression of any part of the gel. The goal is to prevent the strain in the gel from exceeding the threshold of irreversible compression. According to Eq. (62), the limiting strain is qy 1 ezy ¼ ln ð63Þ
: m q0 When m ¼ 3:5, this strain is ezy ¼ 0:29 and qy ¼ 133 kg/m3 . To prevent ez from exceeding ezy , Eq. (56) indicates that the maximum heating rate is dT 1 DH 6 ¼ : dt maF s maF gF L2
Fig. 15. Calculated density at locations corresponding to disks in Fig. 2, when YoungÕs modulus ¼ 0:1 MPa.
ð64Þ
Using Eqs. (52) and (60), with m ¼ 3:5, D ¼ 35 nm2 , L ¼ 2:33 cm, Eq. (64) has been evaluated for several values of H. Fig. 16 shows that the rate used in experiment RSCE B-271 was much too fast, if the modulus of the gel was E ¼ 0:1 MPa (H ¼ 0:11 MPa); on the other hand, if the modulus were initially equal to that of the final aerogel (E ¼ 2:1 MPa), the cycle could have been much faster. Since the density of disk 5 barely exceeded the threshold of irreversible deformation, it is reasonable to infer that the modulus of the gel
G.W. Scherer et al. / Journal of Non-Crystalline Solids 311 (2002) 259–272
Fig. 16. Comparison of thermal cycle used in experiment RSCE B-271 with optimal cycles calculated assuming E ¼ 0:1 or 2.1 MPa.
Fig. 17. Comparison of thermal cycle used in experiment RSCE B-271 with optimal cycle calculated assuming H ¼ 0:5 MPa.
reached about H ¼ 0:5 MPa in the temperature range of maximum strain. As shown in Fig. 17, the optimal cycle is actually shorter than the one used in the experiment, but the rate is slower at low temperatures. More precise predictions would require knowledge of the evolution of H as a function of temperature, and the dependence of permeability on pressure.
6. Summary The major cause of heterogeneity in RSCE aerogels is deformation caused by venting; syneresis can create a denser region near the exterior surface, but it is a relatively minor effect. Measurements of the venting rate for molds containing
271
either pure methanol or a silica gel indicate that the rate of escape is approximately equal to the rate of thermal expansion of the fluid. Using this as a boundary condition, a one-dimensional analysis has been obtained that describes the distribution of pressure and strain within the aerogel in an RSCE mold. The strain is predicted to arise mainly at high temperatures, where the expansion of the fluid increases rapidly. Therefore, the strain of the aerogel can be reduced by decreasing the heating rate in inverse proportion to the product of the viscosity and thermal expansion coefficient of the fluid. A sample calculation shows that this allows reasonable heating rates to be used without causing permanent strain. However, this calculation assumes that the fluid is venting uniformly over the surface of the mold, whereas the original molds vent along a line. Use of a fritted mold has been shown to yield the expected benefit: more uniform strain of the gel, resulting in more uniform final density. Nevertheless, experiment RSCE B-271 produced gels with some densification in the region near the frit. The present analysis shows that the measured densities can be explained by taking account of the low modulus of the initial gel. Using the measured modulus, the model predicts that the gel will be compressed beyond the limit of irreversible strain in precisely that region of the sample where densification was observed. The model permits calculation of an optimal heating cycle (i.e., the highest rate that avoids densification). Using a reasonable estimate of the modulus of the gel at elevated temperature, the optimal cycle is found to be shorter than the one used in RSCE B-271, but uses a slower heating rate at low temperatures. Unfortunately, the project that motivated this study has been cancelled, so we cannot test the predicted optimal cycle using close-loop control of the system. Nevertheless, the experiments to date are in good agreement with the theory, and we expect that highly uniform aerogels could be made by the RSCE process.
Acknowledgements The authors are indebted to Hang-Shing Ma for his help with the beam-bending measurements.
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This work was performed under the auspices of the US Department of Energy by University of California, Lawrence Livermore National Laboratory under contract no. W-7405-ENG-48. That work was described in ÔOptimization of the Rapid Supercritical Extraction Process for AerogelsÕ, UCRL-JC-145444.
References [1] J.F. Poco, P.R. Coronado, R.W. Pekala, L.W. Hrubesh, in: R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton, S.I. Zones (Eds.), Microporous and Mesoporous Materials, MRS Symposium Proceedings, vol. 431, Materials Research Society, Pittsburgh, PA, 1996, p. 297. [2] G.W. Scherer, H. Hdach, J. Phalippou, J. Non-Cryst. Solids 130 (1991) 157. [3] G.W. Scherer, J. Non-Cryst. Solids 145 (1992) 33. [4] G.W. Scherer, J. Sol–Gel Sci. Tech. 3 (1994) 127. [5] T. Woignier, G.W. Scherer, A. Alaoui, J. Sol–Gel Sci. Tech. 3 (1994) 141. [6] J. Gross, P.R. Coronado, L.W. Hrubesh, J. Non-Cryst. Solids 225 (1998) 282. [7] G.W. Scherer, J. Non-Cryst. Solids 142 (1-2) (1992) 18. [8] G.W. Scherer, J. Sol–Gel Sci. Tech. 1 (1994) 169. [9] G.W. Scherer, J. Non-Cryst. Solids 201 (1996) 1. [10] L.W. Hrubesh, C.T. Alviso, in: C.J. Brinker, D.E. Clark, D.R. Ulrich (Eds.), Better Ceramics Through Chemistry III, Materials Research Society Symposium Proceedings,
[11] [12] [13] [14]
[15] [16]
[17] [18] [19] [20] [21]
[22] [23] [24]
vol. 121, Materials Research Society, Pittsburgh, PA, 1988, p. 703. S.-Y. Wang, N.-L. Wu, J. Non-Cryst. Solids 224 (1998) 259. H.N. Holmes, W.E. Kaufmann, H.O. Nicholas, J. Am. Chem. Soc. 41 (1919) 1329. G.W. Scherer, J. Non-Cryst. Solids 108 (1989) 18. G.W. Scherer, C. Alviso, R. Pekala, J. Gross, in: R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton, S.I. Zones (Eds.), Microporous and Mesoporous Materials, MRS Symposium Proceedings, vol. 431, Materials Research Society, Pittsburgh, PA, 1996, p. 497. G.W. Scherer, J. Non-Cryst. Solids 215 (2–3) (1997) 155. R.W. Pekala, L.W. Hrubesh, T.M. Tillotson, C.T. Alviso, J.F. Poco, J.D. LeMay, in: K. Sieradzki, D.J. Green, L.J. Gibson (Eds.), Mechanical Properties of Porous and Cellular Materials, Materials Research Society, Pittsburgh, PA, 1991, p. 197. J. Gross, G. Reichenauer, J. Fricke, J. Phys. D 21 (1988) 1447. G.W. Scherer, J. Non-Cryst. Solids 109 (1989) 171. C.J. Brinker, G.W. Scherer, Sol–Gel Science, Academic Press, New York, 1990 (Chapter 7). J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, 1986. P.E. Liley, T. Makita, Y. Tanaka, in: C.Y. Ho (Ed.), Properties of Inorganic and Organic Fluids, Hemisphere, New York, 1988 (Chapter 20). K.C. Chao, R.A. Greenkorn, Thermodynamics of Fluids, Marcel Dekker, New York, 1975, p. 302. G.W. Scherer, D.M. Smith, X. Qiu, J.M. Anderson, J. Non-Cryst. Solids 186 (1995) 316. D.M. Smith, G.W. Scherer, J.M. Anderson, J. Non-Cryst. Solids 188 (1995) 191.