The concentration of water penetrated into silica depends strongly on the stress state in the surface region. In the water diffusion zone hydroxyl water is generated by the water/silica reaction that is at temperatures <450°C a first-order re action. When [2S ]=[2 SiOH] is the concentration of the im movable hydroxyl, the reaction equation reads
						≡Si-O-Si≡ +H$_2$O ↔ [2 SiOH]
						For this reaction the equilibrium constant is strongly affected by stresses, internal swelling stresses and externally applied stresses. Since no activation volumes for this first- order reaction are available in literature, we will show here the principal influence by a parameter study. As the main results we concluded:
						1) With increasing reaction volume ∆V, the concentration of the hydroxyl S(z) at the surface is significantly reduced and slightly increased inside. 
						2) The shape of the S-distribution deviates strongly from the profile of the total water concentration, C$_w$(z). 
					3) The product of the "effective layer thickness" and the level of the S- concentration, S(0)×z$_{eff}$, is largely independent of the assumed reaction volume

Bucharsky, Claudia

Fett, Theo

Schell, Günter

English

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   KIT SCIENTIFIC WORKING PAPERS Distribution of equilibrium constant k and hydroxyl S in silica surface layers   Theo Fett, Günter Schell, Claudia Bucharsky 195   Institute for Applied Materials  Impressum Karlsruher Institut für Technologie (KIT) www.kit.edu    This document is licensed under the Creative Commons Attribution – Share Alike 4.0 International License (CC BY-SA 4.0): https://creativecommons.org/licenses/by-sa/4.0/deed.en  2022  ISSN: 2194-1629  III      Abstract The concentration of water penetrated into silica depends strongly on the stress state in the surface region. In the water diffusion zone hydroxyl water is generated by the water/silica reaction that is at temperatures <450°C a first-order reaction. When [2S]=[2 SiOH] is the concentration of the immovable hydroxyl, the reaction equation reads  ≡Si-O-Si≡ +H2O ↔  [2 SiOH]  For this reaction the equilibrium constant is strongly affected by stresses, internal swelling stresses and externally applied stresses. Since no activation volumes for this first-order reaction are available in literature, we will show here the principal influence by a parameter study. As the main results we concluded:  1) With increasing reaction volume ∆V, the concentration of the hydroxyl S(z) at the surface is significantly reduced and slightly increased inside.  2) The shape of the S-distribution deviates strongly from the profile of the total water concentration, Cw(z).  3) The product of the "effective layer thickness" and the level of the S-concentration, S(0)×zeff, is largely independent of the assumed reaction volume.    IV                                             V          Contents   1 Basic equations  1 2 Depth profiles  4   2.1  Effect of swelling stresses on hydroxyl concentration for 200°C 4   2.2  Hydroxyl concentrations depending on temperature 7   2.3  Numerical results  7 3  Conclusions  9     References  9                    VI       1 Basic equations Water penetrated into silica reacts with the silica network according to   ≡Si-O-Si≡ +H2O ↔ ≡SiOH+HOSi≡ (1a) with the concentration of the hydroxyl S = [≡SiOH] and that of the molecular water C = [H2O]. The equilibrium constant of this reaction is at temperatures of θ<450°C   CSk =   (2) Water concentrations at silica surfaces under saturation pressure are available from investigations by Öhler and Tomozawa [1] and Zouine et al. [2]. Since the Zouine data extend over the large temperature range of 23°C≤θ≤200°C, these results may be ap-plied in the following considerations. In molar units, the total water concentration is given by    )1( 2121 kCSCCw +=+=  (3) Equations (2) and (3) result in   kCC w211+= ,    (4)     +=kCS w121  (5) By density measurements, it was early shown by Brückner [3, 4], Shackelford [5] and Shelby [6] that the volume of glass increases with increasing water concentration that is due to the high temperatures of ≥1000°C exclusively present in hydroxyl form. The volume expansion strain is proportional to the hydroxyl concentration    Sv ×= κε  (6) In the case of thin water layers at glass surfaces where the glass cannot freely expand, compressive swelling stresses must be generated   )1(3,, νεσσ−−==Evyswzsw  (7) with the hydrostatic stress term σsw,h given as    2   )1(92)( ,,,31, νεσσσσ−−=++=Evzswyswxswhsw  (8) From the work of Le Chatelier [7], it is well known that chemical reactions that exhibit a change in volume will be sensitive to the ambient pressure of the reaction. Changing the pressure, p, changes the equilibrium constant of the reaction and hence the ratio of the concentration of reaction products to reactants. The equation governing the reac-tion is    RTVpk ∆−=∂∂ ln . (9) k is the equilibrium constant, p is pressure, ∆V is the reaction volume, R is the univer-sal gas constant and T is the temperature in °K. The reaction volume ∆V can reach high values [8] for temperatures <450°C and may be different in the temperature regions of <450°C (first-order reaction) and >450°C (second-order reaction).  When [2S]=[2 SiOH] is the concentration of the immovable hydroxyl, the reaction equation (1a) reads  ≡Si-O-Si≡ +H2O ↔  [2 SiOH] (1b) In this case the equilibrium constant for the reaction under hydrostatic pressure is    CS VVV −=∆ ]2[ . (10) with the partial molar volume for [2 SiOH], denoted by ]2[ SV . In a solid the situation is more complicated, since the individual stress components σx, σy, σz are in general independent of each other. The equivalent for the pressure is the hydrostatic stress term, σh= −p, that allows to write eq.(9) as   RTVkh∆=∂∂σln . (11) In the absence of externally applied stresses, it is σh= σsw,h. Then the equilibrium con-stant reads according to [9]      ∆−=RTVSkk λexp0  (12) with λ=18.8 GPa. The hydrostatic stress reads    Shsw λσ −=,  (13)   3 The equilibrium constant as a function of the measured Cw finally results from (12) and (5) in the implicit equation    +∆−=kwCRTVkk1210exp λ  (14) since k is present on the left-hand side as well as on the right-hand side in the denomi-nator of the exponential function.  In the same way, we obtain also an implicit equation for the hydroxyl concentration      ∆+=RTSVkCS wλexp1210 (15) It is obvious that the equilibrium constant varies in a water diffusion zone where Cw varies with depth and an average value k  over the depth profile.   ∫∫==dzzCdzzSCSk)()( (16) The value k is mostly obtained in literature by transmission IR measurements of hy-droxyl and molecular water. These measurements reflect average values k  over the whole water profiles, 0≤z≤W.  The equations (14) and (15) contain two unknown parameters, namely k0 and ∆V which have to be determined by measurements. These values are not freely eligible. They are interrelated by the known value of CS / , see [10]. The experimental results on equilibrium ratios from literature were expressed in [11] for the temperature range of 90°C≤T≤350°C by the empirical relation   −==RTQACSk exp  (17) (A=32.3 and Q=10.75 kJ/mol).  In mass units it is    +=kCS w1211817 (18) (the ratio 17/18 reflects the different mole masses of water and hydroxyl).   4 Figure 1a gives the total water concentration in mass units in a plot with the tempera-ture θ as the linear abscissa. The straight line gives   ( )θ00868.0exp000780.0=wC  (19) Figure 1b illustrates the S-data in a similar plot with a slightly curved averaging curve given by   ( ))/exp(00868.0exp000780.01211817RTQSA+=θ (20) again with the parameters A=32.3 and Q=10750 J/mol   Fig. 1 Total water and hydroxyl data plotted with linear abscissa scaling, a) total water, b) hydroxyl water concentration, curve: eq.(20). 2 Depth profiles  2.1 Effect of swelling stresses on hydroxyl concentration for 200°C Equation (14) makes clear that a depth profile Cw=f(z) of total water must result in a profile of the equilibrium constant k. Very often the measured total water profiles are given by (see e.g. [2])   =bzCC ww 2erfc)0(  (21) 0 50 100 150 200 0.05 0.1 0.2 0.5 S (wt-%) θ    (°C) b) 0 50 100 150 200 0.1 0.15 0.2 0.3 0.5 0.7 θ    (°C) Cw,0 (wt-%) a)   5 In the absence of experimental data for the reaction volume ∆V at temperatures < 450°C, we carried out a parametric study that at least shows the influence of the reac-tion volume in principle. In doing so, we were aware that the reaction volume of the first-order reaction could certainly assume a high value, [8].   Fig. 2 Distribution of the equilibrium constant k in the silica surface for 200°C. First we solved eq.(14) with respect to the equilibrium constant k for 200°C by using the numerical Mathematica-Routine FindRoot [12] under the condition of k =2.1 as is given by eq.(17). The resulting set of curves k=f(z/b, ∆V) are plotted in Fig. 2 for dif-ferently chosen reaction volumes. The “average” equilibrium constant, k =2.1, is indi-cated in Fig. 2 by the red line. This line represents also the equilibrium constant for the case of a disappearing reaction volume, ∆V=0. Then, the hydroxyl concentration profiles S=f(z/b, ∆V) could be obtained from eqs.(18) and (21). Figure 3a shows the results of S normalized on the surface value S0 of the surface hydroxyl water concentration for ∆V =0 and a temperature of 200°C. Figure 3b additionally shows the results for 100°C (blue curves) where the boundary condition now reads k =1.1. The dashing of the curves corresponds to the same reaction volumes as in the case of 200°C.  All results show a strong variation of the local equilibrium constant k(z). 500 z/b 1 2 3 4 200° 100 50 10 5 1 k 30 60 90 120 180 300 400 500 ∆V  (cm3/mol) k=2.1   6  Fig. 3 Distribution of hydroxyl concentration S in the silica surface, a) for 200°C and a wide range of the activation volume, b) comparison of results for 100° and 200°C.    Fig. 4 Effect of swelling stresses on hydroxyl concentration at the surface (left ordinate) and the relat-ed hydrostatic and linear swelling stresses (right ordinates). 60 120 180 500 0.05 0.1 0.2 0.5 S (wt-%) 100 150 200 θ    (°C) 0 ∆V  (cm3/mol) 100 40 20 -σh (MPa) 60 150 30 -σx, -σx (MPa) 300 1 2 3 4 z/b ∆V  (cm3/mol) 0 30 60 90 120 180 200°C 1.0 0.8 0.6 0.4 0.2 300 400 500   S S0,200°C a) 1.0 0.8 0.6 0.4 0.2 1 2 3 4 z/b ∆V  (cm3/mol) 0 30 60 90 120 180 100° 200° k=2.1 k=1   S S0,200°C b)   7 2.2 Hydroxyl concentrations depending on temperature The data of Fig.1 b represented by the solid line for ∆V=0 are plotted in Fig. 4 together with the curves for a number of reaction volumes. Whereas the left ordinate shows the hydroxyl concentrations at the surface, the right ordinates represent the related hydro-static and equi-biaxial swelling stresses according to eqs.(7) and (8). A clear decrease of the stresses is visible for an increasing reaction volume. The S-curves, strongly de-viating from the erfc-shaped total water distribution, give rise for an effective layer thickness beff as is defined by the depth in which the concentrations are reduced to 50% of the surface values.  2.3 Numerical results Figure 5 summarizes the results of the numerical computations. From the example in Fig. 5a it can be seen, that the layer width increases by a factor of about 2 with ∆V in-creasing from 0 to 500 cm3/mol. Simultaneously, the surface S-concentration and the swelling stresses decreased as is visible from Fig. 4. Figure 5b shows the surface S-concentration at 200°C and Fig. 5c the related layer thicknesses beff, both depending on the reaction volume ∆V. Finally, Fig. 5d gives the product beff×zeff as a result of the reaction volume. This quantity also represents in principle the mechanical bending moment that is caused by the swelling stresses. The solid line indicates the average value over the wide region of parameter ∆V and the dash-dotted lines show ±1 SD (SD=Standard Deviation).     1.0 0.8 0.6 0.4 0.2 1 2 3 4 z/b ∆V=0  (cm3/mol)   S Cw,0,200°C a) ∆V=500  (cm3/mol) b zeff ∆V  (cm3/mol) 0.4 0.35 0.3 0.25 100 200 300 400 500 S (wt-%) b)   8   Fig. 5 Influence of swelling on the maximum concentrations and half-width beff of the profiles, a) definition of an effective layer thickness zeff, b) S-concentration at the surface z=0, c) effective zone thickness, d) product of surface concentration and effective layer width, representing bending moment due to water uptake.  Fig. 6 Numerical dependency between the parameters k0 and ∆V for 200°C 500 500 100 50 10 5 400 300 200 100 ∆V  (cm3/mol) k0 ∆V  (cm3/mol) 100 200 300 400 500 0.005 0.004 0.003 0.002 0.001 d) S×zeff/b 2.0 ∆V  (cm3/mol) 100 200 300 400 1.5 1.0 0.5 c) zeff/b   9 Finally, Fig. 6 shows the numerically obtained dependency between the parameters k0 and ∆V. For ∆V > 60 cm3/mol it can be written   ,]exp[ 210 VAAk ∆≅  (22) with A1=2.337, A2=0.010485 mol/cm3, represented by the solid straight line in Fig. 6. 3 Conclusions The main results of the present parameter study on the influence of the swelling stress-es on the depth profiles of the hydroxyl concentration S(z) can be summarized as fol-lows:  1) With increasing reaction volume ∆V, the concentration of the hydroxyl S(z) at the surface, z=0, is significantly reduced and slightly increased inside (for z>b).  2) The shape of the S-distribution deviates strongly from the profile of the total water concentration, Cw(z).  3) The product of the "effective layer thickness" and the level of the S-concentration, S(0)×zeff, is largely independent of the assumed reaction volume. This influence has the potential to interpret the experimental finding of agreement between water ab-sorption and bending moment as well as the relatively low absolute value of the threshold stresses at the surface [11]. This will be shown in a separate report. References                                                  1 Oehler, A., Tomozawa, M., Water diffusion into silica glass at a low temperature under high water vapor pressure, J. Non-Cryst. Sol. 347(2004) 211-219. 2 A. Zouine, O. Dersch, G. Walter and F. Rauch, “Diffusivity and solubility of water in silica glass in the temperature range 23-200°C,” Phys. Chem. Glass: Eur. J. Glass Sci and Tech. Pt. B, 48 [2] 85-91 (2007). 3 Brückner, R., “The structure-modifying influence of the hydroxyl content of vitreous sili-cas,” Glastech. Ber. 43 (1970), 8-12. 4 Brückner, R., “Metastable equilibrium density of hydroxyl-free synthetic vitreous silica,”  J. Non-Cryst. Solids, 5 (1971), 281-5. 5 Shackelford, J.F., Masaryk, J.S., Fulrath, R.M., “Water Content, Fictive Temperature and Density Relations for Fused Silica,” J. Am. Ceram. Soc. 53 (1970), 417. 6 Shelby, J.E., “Density of vitreous silica,” J. Non-Cryst. 349 (2004), 331-336. 7 H. Le Chatelier, C.R. Acad. Sci. Paris 99(1884), 786. 8 G. Schell, G. Rizzi, T. Fett, Hydroxyl concentrations on crack surfaces from measurements by Tomozawa, Han and Lanford, 82, 2018, ISSN: 2194-1629, Karlsruhe, KIT.   10  9 T. Fett, G. Rizzi, K.G. Schell, C.E. Bucharsky, P. Hettich, S. Wagner, M. J. Hoffmann, Consequences of hydroxyl generation by the silica/water reaction, Part I: Diffusion and Swell-ing, KIT Scientific Publishing, 101(2022), Karlsruhe.  10 T. Fett, G. Schell, Elimination of swelling stresses from measurements of the equilibrium constant in silica, 117, 2019, ISSN: 2194-1629, Karlsruhe, KIT 11 S. M. Wiederhorn, F. Yi, D. LaVan, T. Fett, M.J. Hoffmann, Volume Expansion caused by Water Penetration into Silica Glass, J. Am. Ceram. Soc. 98 (2015), 78-87. 12 Mathematica, Wolfram Research, Champaign, USA.                                11      KIT Scientific Working Papers ISSN 2194-1629www.kit.eduKIT – The Research University in the Helmholtz Association

Distribution of equilibrium constant k and hydroxyl S in silica surface layers

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Distribution of equilibrium constant k and hydroxyl S in silica surface layers

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