- Research
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# Acid polishing of lead glass

- Jonathan A Ward
^{1}, - Andrew C Fowler
^{1}and - Stephen BG O’Brien
^{1}Email author

**1**:1

https://doi.org/10.1186/2190-5983-1-1

© Ward et al.; licensee Springer 2011

**Received:**11 November 2010**Accepted:**3 June 2011**Published:**3 June 2011

## Abstract

### Purpose

The polishing of cut lead glass crystal is effected through the dowsing of the glass in a mixture of two separate acids, which between them etch the surface and as a result cause it to be become smooth. In order to characterise the resultant polishing the rate of surface etching must be known, but when this involves multicomponent surface reactions it becomes unclear what this rate actually is.

### Methods

We develop a differential equation based discrete model to determine the effective etching rate by means of an atomic scale model of the etching process.

### Results

We calculate the etching rate numerically and provide an approximate asymptotic estimate.

### Conclusions

The natural extension of this work would be to develop a continuum advection-diffusion model.

## Keywords

- Etching rate multi-component
- crystal glass
- mathematical model
- ordinary differential equation
- asymptotics
- numerics
- Laplace transform

## 1 Introduction

Wet chemical etching, or chemical milling, is a technique which allows the removal of material from a substrate via chemical reaction. In many applications selective attack by the chemical etchant on different areas of the substrate is controlled by removable layers of masking material or by partial immersion in the etchant. Etching is used in a wide variety of industrial applications, from the manufacture of integrated circuits to the fabrication of glass microfluidic devices [1–4]. Stevens [5] gives a qualitative description of etching in the context of tool design, masks for television tubes and fine structures in microelectronics. More recent accounts are to be found in [6, 7], who also include a detailed examination of the chemistry of these processes. In this paper we wish to examine the etching of rough lead crystal glass fully immersed in a bath of acid etchant. The evolution of the surface is determined by the rate of the surface reaction which dissolves the solid surface. For a simple reaction involving a single solvent and a monominerallic surface, this rate is simply characterised by the reaction rate kinetics. However, if more than one solvent is necessary to dissolve a surface with several different components, it is not clear what the effective surface dissolution rate should be.

This paper is concerned with this latter situation, and is motivated by *multicomponent* etching used in the production of lead crystal glassware. This problem was introduced at a European study group for industry at the university of Limerick in 2008 (ESGI 62). In this case, decorative features cut into the glass leave it optically opaque and polishing is subsequently required to restore its transparency [1]. The process consists of sequential immersion of the glass in a mixture of hydrofluoric (HF) and sulphuric (H_{2}SO_{4}) acid, followed by rinsing to remove insoluble lead sulphate particles from the interface. The etching and rinsing steps are repeated a number of times. In particular, we focus on the wet chemical etching step, where it is necessary to use both hydrofluoric and sulphuric acid in order to dissolve all of the components of the glass, namely SiO_{2}, PbO and K_{2}O. The potassium salts and silicon tetrafluoride are soluble whereas the lead sulphate is not, hence the required rinsing. Such multicomponent systems pose a non-trivial problem in determining the consequent etching rate [8].

There is a large literature concerning experimental studies of wet chemical etching of glass; see [1], for a review. These studies are primarily concerned with the measurement of etching rates [2] and how these are related to different etchant and glass compositions [3, 9, 10]. It has been shown that etching rates of multicomponent glasses, where all the components dissolve in HF, have a non-trivial dependence on both the ‘bonding connectivity’ of the glass and the presence of reaction by-products on its surface [10]. To our knowledge, there has not been an experimental study of multicomponent glasses where different types of etchant are required. An observation related to the work in this paper is that etching of a smooth surface (for example, one which has been mechanically polished) causes it to develop cusp like features [1, 3], thus roughening it slightly. The height of such features is found to be normally distributed [11].

**v**satisfies ${F}_{t}+\mathbf{v}.\mathbf{\nabla}F=0$, whence also ${F}_{t}+{v}_{p}|\mathbf{\nabla}F|=0$, where ${v}_{p}=\mathbf{v}.\mathbf{n}$ denotes the normal velocity of the surface, and $\mathbf{n}=\frac{\mathbf{\nabla}F}{|\mathbf{\nabla}F|}$ is the unit normal. For example, if the surface is denoted by $z=s(x,y,t)$, then (taking $F=s-z$)

*κ*is defined by

where ${v}_{p}$ is an increasing function of the curvature *κ*: a first approximation might take the form ${v}_{p}(\kappa )={v}_{p0}(1+\alpha \kappa )$. Hence (1.3) is a non-linear diffusion equation for *s*. As such, the surface will smooth as it is etched, and this would explain simply enough why polishing works. From an experimental point of view, halting the etching processes at various times and examining the surface microscopically is an obvious way of testing the validity of the above mechanisms. The latter mechanism will generally give surfaces which are progressively smoother while the former (constant ${v}_{p}$) could lead to the development of intermediate cusps prior to the ultimate removal of asperities.

where **v** is the etching rate, ${\sigma}_{e}$ is a constant and *c* is the etchant concentration. He then refined this model to deal with the case of a mask with a finite hole [18]. Later, this approach was further developed and successfully compared with experiment [19]. While this might suggest, at least in the system they considered, that the dependence of etching rate on surface energy is weak, it should also be noted that the substrate to be etched was initially smooth. The etching of lead glass is different in that the initial substrate has a rough surface of high curvature.

_{2}SO

_{4}). Lead crystal consists largely of lead oxide PbO, potassium oxide K

_{2}O, and silica SiO

_{2}, and these react with the acids according to the reactions

The surface where the reaction occurs is a source for the substances on the right hand side and a sink for those on the left hand side. The potassium salts are soluble, as is the silicon hexafluoride, but the lead sulphate is insoluble and precipitates on the cut surface, from which it is washed away in the rinsing bath. In fact this rinsing action must be chemical, with the water acting to dissolve the bonds which tie the sulphate crystals to the surface.

Spierings [1] points out that the mechanism of the etching reaction is not well understood at molecular level: our aim in this paper is to elaborate upon previous work [8] where we proposed a *microscopic* model to capture the salient features of multicomponent etching, with the aim of determining the effective etching rate.

The outline of the paper is as follows. In the section ‘Modelling multicomponent etching’ we discuss the mechanical process of etching, in particular where more than one solvent is necessary, and we indicate a conundrum which arises in this case. We then build a model which describes the evolution of the surface at an atomic scale, describing in particular the evolution of atomic scale surface roughness. This model is solved in the section ‘Solution of the discrete model’, and the resulting effective etching rate is determined. A feature of the solution is that, although the model describes the evolution of a site occupation density on a discrete lattice, the numerical solutions strongly suggest that a continuum approximation should be appropriate. In the section ‘A continuum model’, we derive such a model and study its solutions. Surprisingly, we find that the consequent etching rate differs from that computed from the discrete model, and we offer an explanation for why this should be so. The conclusions follow in the last section.

## 2 Modelling multicomponent etching

### 2.1 Etching rate

*j*in the glass is ${\varphi}_{j}{\rho}_{j}$, and its molar density (moles per unit volume) is

*j*disappears from the surface is $\frac{{\varphi}_{j}{\rho}_{j}{v}_{p}}{{M}_{j}}$, and this must be equal to the rate of disappearance ${R}_{j}$ for each species in the glass, measured in moles per unit area of surface per unit time. (Thus ${R}_{j}$ has the units of a molar flux.) Hence

*j*. Assuming that there is always an excess of acid available for reaction with the three species in the glass, it is natural to assume the balance

*j*, and thus (2.2) and (2.6) imply

While this is a statement that the flux of acid to the surface exactly balances the ‘flux’ of surface disappearing via chemical reaction, it leads us to what we will call the Tocher conundrum. (This observation was made by Dave Tocher during ESGI 62 at the University of Limerick.) The mathematical part of this conundrum lies in the general impossibility of satisfying (2.7) for each species, since it would require the specific effective reaction rates ${F}_{i}$ to be related to each other, and this is unrealistic. In order to determine what the etching rate ${v}_{p}$ is, we thus need to consider in greater detail just what the surface reaction process is.

_{2}SO

_{4}, by means of the following conceptual picture. Imagine the glass as a crystal lattice (this is not actually the case, being a glass, but the concept is valid), where lead sulphate, silica and potassium oxide molecules are distributed at random. The sulphuric acid can pick off the lead oxide molecules, and we suppose that it can excavate downwards into the lattice until it encounters a silica molecule. At this point, no further stripping is possible, and reaction at that horizontal location ceases. This stripping will happen at each point of the surface, and, supposing only vertical excavation is possible, eventually a molecularly rough surface will be obtained, in which only silicon molecules are exposed, thus preventing any further reaction. This process is represented qualitatively in Figure 2.

### 2.2 Microscopic model development

In order to describe the surface reaction, we need to account for the molecularly rough surface, and to do this, we again suppose that the molecules are arranged in a lattice, with the horizontal layers denoted by an index *n*, with $n=0$ indicating the initial surface, and *n* increasing with depth into the lattice. As etching proceeds, the surface will have exposed sites at different levels. We let ${\psi}_{n}^{j}$ denote the fraction of exposed surface at level *n* of species *j*.

To clarify this, let us assume there are *M* sites in the horizontal and $N+1$ rows in the vertical (see Figure 2) so that $n=0,\dots ,N$. Then ${\psi}_{n}^{j}$, at any level or row, *n*, is the number of exposed sites of type *j* divided by *M*.

*j*is denoted ${F}_{j}$, and the species is present in a fraction of sites ${f}_{j}$ in the crystal (that is, ${f}_{j}=$ number of

*j*molecules divided by $M(N+1)$). Thus

*n*(that is, the number of exposed sites at level

*n*divided by the total number of sites

*N*in any row). For example, we illustrate a three species case in Figure 2. At $t=T$, that is, in the middle section of Figure 2, we see that

with all other ${\psi}_{n}^{j}$ being zero.

*j*sites, while the positive term represents the creation of new exposed sites at level

*n*(a fraction ${f}_{j}$ of which are

*j*sites) as sites at level $n-1$ are etched away. The initial conditions are simply:

*j*sites) to replicate an impenetrable substrate. Thus for simplicity, and mindful of the fact that each level represents a layer of molecules, we consider the glass to be infinitely deep in effect. Note that (2.11) and (2.12) imply the conservation law:

*j*molecules. Thus, for example, ${A}_{S}=0$ if the acid is H

_{2}SO

_{4}(which does not break down SiO

_{2}molecules, see (1.5)). The ${A}_{j}$ (units s

^{−1}) are given by

*N*is Avogadro’s number ($6\times {10}^{23}$ mole

^{−1}), and Δ

*x*is the lattice spacing (m). Note that the molar density is

unless the ${A}_{j}$’s are equal.

## 3 Solution of the discrete model

### 3.1 Numerical solution

*n*into the crystal of type

*j*, is positive) is diffuse (that is, it spreads out as it moves down into the crystal), and propagates downwards at an essentially constant rate. Note also that the discrete solution appears to be well approximated by a continuously varying site occupation density for each species.

### 3.2 Analysis of the discrete model

*s*,

*s*,

*j*are ordered in terms of increasing reaction rate so that ${A}_{1}<{A}_{2}<\cdots $; it then follows that at large

*t*,

*n*. Since (3.13) implies that ${G}_{1}>0$, it follows that there are two cases to consider. If ${G}_{1}<1$ then solutions decay in both

*n*and

*t*, but this tells us nothing about the mean etching rate. When ${G}_{1}>1$, (3.15) indicates that ${\psi}_{n}^{j}$ increases with

*n*; however, this asymptotic result must become inappropriate when $n\sim t$, since conservation of sites implies

as in (2.13). Thus ${\psi}_{n}^{j}$ is bounded and in fact decreases towards zero at large *n*. We wish to focus attention on the penetration depth, or wavefront location, of the etchant so we will denote its location by ${n}_{w}(t)$.

#### 3.2.1 Solutions for large *n* and *t*

*t*with ${v}_{p}$ and

*ξ*fixed. In particular, we will choose ${v}_{p}$ to be the speed of the wavefront. We define

*J*species. The integrand of (3.19) has poles at $-{A}_{1},-{A}_{2},\dots ,-{A}_{J}$, at which

*g*is infinite; between these values,

*g*is monotonically decreasing, and therefore

*g*has $J-1$ real zeroes at ${\lambda}_{1},\dots ,{\lambda}_{J-1}$, where $-{A}_{j}>{\lambda}_{j}>-{A}_{j+1}$. It follows from this that

*g*is convex and decreasing for $\lambda >-{A}_{1}$, it follows that

*ρ*is real and convex for $\lambda >-{A}_{1}$, with a unique minimum at $\lambda ={\lambda}^{\ast}$, say. The form of Re

*ρ*as a function of

*λ*is shown in Figure 5.

To evaluate the integral (3.19) asymptotically for large *t*, we aim to deform the contour $\Gamma =[\gamma -i\infty ,\gamma +i\infty ]$ to one passing through a saddle point of $\rho (\lambda )$, where ${\rho}^{\prime}(\lambda )=0$, since the size of the exponent in the integrand is dominated by *ρ*. We can then use the method of steepest descents. The obvious such saddle point is at ${\lambda}^{\ast}$, and this is indeed the correct choice. To understand why, we need to describe the steepest ascent and descent paths in the complex *λ* plane.

These are given by the curves $Im\rho =$ constant, which trace out trajectories in the complex *λ* ‘phase plane’. (If we think of *ρ* as a complex velocity potential, then the curves $Im\rho =$ constant are the streamlines.)

*ρ*on these curves, the points $-{A}_{j}$ are like sinks ($Re\rho \to -\infty $), and the points ${\lambda}_{k}$ are like sources ($Re\rho \to \infty $). On the real axis, the source lines from $(-\infty ,{\lambda}_{J-1},\dots ,{\lambda}_{1},{\lambda}^{\ast})$ are directed to the sinks at $-{A}_{j}$, as shown in Figure 6. Nearby trajectories (of constant Im

*ρ*) must do the same, passing to the sinks on either side; hence there must be a dividing trajectory which must go to infinity.

However, $\rho \sim \lambda $ at infinity, and it follows from this that firstly, the steepest descent trajectories from ${\lambda}^{\ast}$ asymptote to ∞ horizontally in the left hand part of the plane, and consequently so must also the lines of constant Im*ρ* which reach infinity from the sinks at $-{A}_{j}$. Most of the streamlines from the sources terminate on the adjoining sinks, but the dividing streamline, being sandwiched between adjoining trajectories which originate at $Re\lambda =-\infty $, must also originate there. But this is only possible if Re*ρ* decreases, and this requires that each dividing streamline from the $J-1$ sources passes through a saddle point, as shown in Figure 6. Since *ρ* has real-valued coefficients, the phase plane is symmetric about the real axis, so that there are another $J-1$ saddles in the lower half plane.

### 3.3 Summary of solutions

*j*at depth

*n*into the crystal is given by (3.14), incorporating (3.12) and (3.13). For large times, this can be simplified to (3.15) which is inappropriate when $n\sim t$. For large

*t*and $n\sim t$, we then found that the asymptotic limit giving the key result for the penetration ${n}_{w}$ of the wavefront into the crystal, at time

*t*, as

where ${f}_{k}$, ${A}_{k}$ are the initial fractions and reaction rates of the species *k* in the solid. The corresponding asymptotic result for ${\psi}_{n}^{j}$ is given by (3.26).

In the context of Figure 4, the asymptotic solution (3.24) in conjunction with (3.17) predicts that the interface, neglecting its diffusion, moves at a speed ${v}_{p}={(\frac{{f}_{1}}{{A}_{1}}+\frac{{f}_{2}}{{A}_{2}})}^{-1}=1.54$, that is, ${n}_{w}\sim 1.54t$. It is apparent that the individual etchant rates sum like electrical resistors in parallel. The fact that the numerical solutions in Figure 4 are so smooth, and that they are so close to the asymptotic solution, suggests strongly that a continuum model should be appropriate. We now examine this possibility.

## 4 A continuum model

*ϕ*satisfies

where *ρ* is defined in (3.25).

It is thus natural to suppose that this result can be found easily and more simply than the earlier discrete calculation, and that it would provide a more suitable vehicle for further development of the model, for example, in considering the shape of the interface on the reaction rates. Consequently, it is surprising to find that a simple continuous approximation apparently fails to reproduce the exact, discrete result.

*B*. By inspection, one eigenvalue and eigenvector pair of

*B*is

**v**(to see this, write

**v**as a linear combination of

**f**and a vector orthogonal to

**a**; or simply do the calculation explicitly). In particular,

in agreement with (3.6).

*x*, thus ${e}^{-{\partial}_{x}}\approx 1-{\partial}_{x}+\frac{1}{2}{\partial}_{xx}$, this becomes

be satisfied.

where $\mathit{\psi}={({\psi}^{1},\dots ,{\psi}^{J})}^{T}$, and bears a suggestive resemblance to (4.3).

*ϕ*in approximating (4.14) by (4.15). In a similar way, the ansatz $\mathit{\psi}\sim {v}_{p}{A}^{-1}\varphi \mathbf{f}$ in (4.18) leads to the inconsistent equation

It seems that only the discrete formulation gives a consistent description of the solution. Nevertheless, the simple form of (4.2) and (4.3) suggests that a derivation of an appropriate advection-diffusion equation should be possible, but it is opaque as to how to do this.

## 5 Conclusions

*j*in the solid, while ${A}_{j}$ is the reaction rate (rate of surface removal) of each species if present on its own. This thus gives an approximation for the overall etching rate, neglecting its diffusion.

that is, this predicts that ${n}_{w}\sim 1.54t$ in Figure 4 where the wavefront ${n}_{w}$ is located at the centre of each of the Gaussian-like curves. There is obvious good agreement. Within the terms of the model we propose, this shows that the solid behaves as if it were layered, with the layers of each species being parallel to the surface, so that the overall rate is determined by the weighted sum of the inverse rates. The constituent etching rates thus sum like electrical resistors in parallel. This surprising conclusion is not at all intuitive, and shows the importance of providing an adequate model for the process.

In addition, we have shown that, although the discrete lattice model has numerical solutions which are smooth at large times, the apparently simple expedient of Taylor expanding the variables and truncating the resulting expansion simply leads to the wrong result. It remains unclear why this should be so, or what the correct averaging method should be to derive an appropriate continuous approximation.

## Declarations

### Acknowledgements

We acknowledge the support of the Mathematics Applications Consortium for Science and Industry (http://www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative grant 06/MI/005, the Stokes grant 07/SK/I1190 and the PI grant 09/IN.1/I2645.

## Authors’ Affiliations

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## Copyright

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