SILVER-MIRRORING EDGE PATTERNS: DIFFUSION-REACTION MODELS FOR THE FORMATION OF SILVER MIRRORING ON SILVER GELATIN GLASS PLATES
GIOVANNA DI PIETRO, & FRANK LIGTERINK
In this appendix an equation is derived to describe the diffusion of a gas between two reactive surfaces. The law of mass conservation states that, for a steady state concentration profile, the difference between the amount of gas leaving and entering a small air volume of dimensions dx � h � a (cm3) (fig. 15) per unit time [(dmout/dt) – (dmin/dt)] (g s-1) is equal to the amount of gas removed from the air volume because of reaction with the emulsion-dmr/dt (g s-1):
At equilibrium, the amount of gas entering a small volume between the plates is equal to the amount of gas leaving plus the amount of gas reacting with the emulsion. The two plates are taken as being parallel and with emulsion sides facing.
The left-hand part of equation (A1.1) is found by considering that the mass of gas entering through an area h � a (cm2) is the gas flow times the area and by applying the first law of diffusion, Fick's law (Crank 1975):
where c(x) (g cm-3) is the gas concentration and D(cm2 s-1) is the gas diffusion constant.
The right part of equation (A1.1) is found with the reaction collision theory (Volpe and Peterson 1989; Atkins 1990), that is, by calculating the number of collisions between the gas and the top and bottom emulsion surfaces per unit time and assuming that only a fraction γ of them will give rise to a reaction:
R is the gas constant (R = 8.313�107 g cm2 s-2K-1 mol-1), T (K) the absolute temperature, and M (g mol-1) the gas molecular weight. By substituting equations (A1.3) and (A1.2) in (A1.1) and simplifying, we obtain the final equation describing the gas concentration between the plates at equilibrium:
with boundary conditions of constant gas concentration c0 just outside the plates at x = 0.
The problem (A1.4) has a simple exponential solution:
The amount of mirroring on the plate will be proportional to the amount of gas reacting with the plate per unit surface and time, the so-called gas absorption rate Ψ (g cm-2 s-1). The gas absorption rate is found by substituting the concentration profile equation (A1.5) in equation (A1.3) and dividing by the total surface area of bottom and top emulsion surfaces (2a � dx):
With the hypothesis of high reactivity and no ventilation, the steady-state diffusion equation (Crank 1975) for the gas concentration c (g cm-3) surrounding a freely exposed plate is:
with boundary conditions
where c0 (g cm-3) is the gas environmental concentration.
The amount of gas reacting per unit surface and unit time with the plate, the so-called gas absorption rate Ψ (g cm-2 s-1), can be calculated from the concentration profile solution of (A2.1)–(A2.2) with Fick's law (Crank 1975) being equal to the gas flow J (g cm-2 s-1) calculated at the plate surface:
where D (cm2 s-1) is the gas diffusion constant.
Problems (A2.1)–(A2.2) can be solved analytically in the case of a plate with circular shape (Landau and Lifshitz 1960, 27).
The problem solved in the reference is finding the electrostatic potential Φ generated by a charged disk of radius a. This problem is identical to the gas diffusion problem (A2.2) provided the following substitutions are performed: Φ → (c0-c), e → c0�2a/π, ς → Ψ/(4πD), where e is the electric charge and ς the surface charge density on the plate. The solution presented in the reference for the charge distribution differs from the present solution by a factor of 2, since in a glass plate negative only one face is covered with emulsion and reacts with the gas.
The gas absorption rate is in this case given by:
where a (cm) is the radius of the plate and D (cm2 s-1) the gas diffusion constant in air.
Figure 8a shows the contour plot of the absorption rate (A2.4).
In the more realistic case of a rectangular plate, problem (A2.1)–(A2.2) cannot be solved analytically. We solved the problem with the finite element analysis numerical technique. The finite element method consists of dividing the space into small areas and looking for approximate solutions in each of them. The solutions are then combined to get an approximate solution on the whole space. As this method requires a large number of calculations, it is normally performed with the aid of computer programs. We used a freeware program available on the Web called Tochnog (www.tochnog.com). The solution for a 9 � 12 cm plate was found within a three-dimensional square box with dimensions of 50 � 50 � 50 cm. The box was divided into regular cells by drawing a grid of 1.5 cm spacing. The plate resulted as being covered by 77 nodes. The program calculates the value of the gas concentration and of the gas flow at each node. The values of the gas flow at the nodes on the plate give the gas absorption rate. Figure 8b shows the contour plot of the absorption rate for the rectangular plate.
In order to evaluate the buckling of a glass plate under the action of a shrinking layer of gelatin, we will analyze a simplified model. Let us consider a glass bar of length L (cm) equal to the longest side of the glass plate and of rectangular cross section of dimensions dg � δ (cm2), where dg (cm) is the thickness of the glass plate and δ is an arbitrary length with δ<<L. A gelatin layer of cross section of dimensions de � δ (cm2), where de (cm) is the thickness of the emulsion, covers the bar. The gelatin layer is taken as being attached to the glass only at the two extreme ends of the bar.
During shrinking, the gelatin exerts a force directed toward the center of the bar and applied at its two ends. This force is given by the stress ς (g cm-1 s-2) applied by the gelatin times the gelatin cross section:
The theory of the buckling of beams (Feynman et al. 1964) allows for the prediction of the minimum force applied at the two ends of the bar necessary to deform it, the so-called Euler force:
where Y (g cm-1 s-2) is the Young's modulus of the glass and I (cm4) is the moment of inertia of the bar cross section:
Substituting (A3.3) in (A3.2) and equating (A3.1) and (A3.2), we get an expression for the minimum stress able to deform the bar:
Taking L as the largest side of the glass plate, L = 12 cm, dg = 0.13 cm, de = 5 � 10-3 cm, and Y = 3 � 1011 to 6 � 1011 g cm-1 s-2, depending on the glass composition (Uhlmann 1980), we get ς = 7.5 � 108to 1.5 � 109 g cm-1 s-2. This value can be compared with the stress developed by a gelatin layer when drying from 80% RH to 20% RH, ς = 4000 PSI (Mecklenburg et al. 1994) equivalent to 2.8 � 107 g cm-1 s-2.
The minimum stress necessary to buckle the glass bar is 20 to 50 times greater than the stress actually exerted by a layer of drying gelatin. No data are available in the literature for the stress developed by a gelatin layer during aging.
The authors would like to thank U. B. Kejser for providing many references on silver mirroring, S. Dobrusskin for providing the plates shown in figures 1 and 3, and J. S. Johnsen for providing the recipe modified from the standard hydrogen peroxide test.
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SOURCES OF MATERIALSGlass plate negatives: Agfa APX Avipan 100 (90�120�1.5 mm)
Agfa-Gevaert AG Stettbachstrasse 7 CH-8600 D�bendorf Switzerland
Fluka Chemie AG Postfach 260 CH-9471 Buchs SGI Switzerland
GIOVANNA DI PIETRO studied physics at the University of Milan, where in 1994 she received her master's degree with a specialization in experimental physics. Since 1996 she has collaborated with the Instituut Collectie Nederland as a conservation scientist. In June 2002 she received her Ph.D. in chemistry at the University of Basel's Laboratory of Scientific Photography with a thesis on silver mirroring on silver gelatin glass plates. Address: Laboratory of Scientific Photography, University of Basel, Klingelbergstrasse 80, CH-4056, Switzerland.
FRANK LIGTERINK, a paper conservator, graduated in 1989 from the Opleiding Restauratoren, Amsterdam. From 1989 to 1995 he studied at the Vrije Universiteit, Amsterdam, where in 1995 he received his master's degree in physics with a specialization in theoretical physics. In 1996 he received his qualification as a physics teacher. Currently he is working at the Instituut Collectie Nederland as a conservation scientist. Address: Instituut Collectie Nederland, Gabriel Metsustraat 8, 1071 EA Amsterdam, The Netherlands.