## 3 Pinhole Models

### Quick Start

The presence of a pinhole in an otherwise perfect barrier would seem to give an obvious increase in permeation of water or oxygen. In fact it is far from simple. This is a tutorial app describing three possible models, with the third being the most realistic and surprising. If you are serious about good barriers for foods, pharma or printed electronics, then knowing these models, and when to apply them, is vital.

A famous paper by Decker & Henry1 addresses some of the real-world issues that affect high-quality thin-film barriers. These are typically those created via vacuum processes, but the ideas are fully generalisable. They define 3 models:

1. Electrical Model - As if each layer had a resistance R that can be summed via the "1/R" rule as with electrical resistors
2. Coverage Model - Although the barrier material is perfect its coverage is not perfect so the barrier is less good than it should be
3. Pinhole Model - Although the barrier material is perfect, it contains pinholes through which molecules can diffuse

In the examples that follow only two layers will be modelled, though the ideas can obviously be extended to an arbitrary number of layers. The first layer (S) is the substrate and the second layer (B) is the barrier film. We want to know the Barrier Improvement Factor, BIF, from adding the barrier. The units of the Transmission Rate are deliberately left un-stated as we are interested in relative, not absolute, effects.

### Electrical Model

The Permeation app has already described the Electrical Model. Given the layers with Transmission Rates TRS and TRB, the total transmission rate and BIF are given by:

1/TRtot=1/TRS+1/TRB, BIF=1+TRS/TRB

 TRS TRB TRtot BIF

### Coverage Model

This simply assumes that the barrier layer is perfect but contains a fraction Θ of defects varying from 0 (perfect barrier layer) to 1 (no barrier layer). The equations are simple:

TRtot=ΘTRS; BIF=1/Θ

 TRS Θ TRtot BIF

The Coverage model is quite comforting for those who want to believe that small amounts of defects are insignificant in terms of reducing barrier performance. For thin substrates it is valid. But for thick substrates it is completely misleading, and the Pinhole model is required.

### Pinhole Model

This assumes that the barrier layer is perfect and contains Φ pinholes per unit area and each pinhole has a radius r. The substrate has a thickness d. It turns out that the ratio λ=d/r has a significant effect on the permeation. The area fraction Θ of defects is given by πr²Φ (slightly modified from the original via the definition of Θ) and the equations are:

TRtot=ΘTRS(1+1.18λ), BIF=1/Θ(1+1.18λ)

When the pinholes are large compared to the substrate thickness (d/r << 1 so λ << 1) then the formula is identical to the Coverage formula. Because the formula explores the effect of the thickness of the film, the input is TSS10, i.e. the Transmission Rate for a 10μm film.

 TRS10 Φ /m² r μm d μm Θ TRtot BIF

The pinhole model makes it clear that the most important thing for a good barrier is cleanliness and lack of pinholes. The best barrier technology in the world is going to be undermined by just a few pinholes. We all think we know this, but the calculation brings home just how important it is to control the number and size of pinholes. Although increasing the thickness of the substrate helps reduce the TRS value, it increases (via the d/r term) the overall effect of the pinholes so one has to be careful about the choice of substrate thickness.

Once d/r >> 1 there is no net effect of increasing d because TRS~1/d. Thus there is a critical thickness above which the substrate thickness has no effect on permeation. This can be used to great advantage. A relatively thin coating of an expensive barrier on top of a relatively thick film of a poor barrier gives the advantages of the poor barrier (e.g. physical robustness of the thick film and low cost) with the barrier properties of the expensive coating.

Why is the d/r term significant? Because molecules can diffuse sideways and not just in the direction straight through the hole. Although sideways diffusion is slower (because the molecule has to travel a greater distance) there are (for thick substrates) many more paths to travel, so the net effect is a significant boost. For thin substrates, the extra diffusion paths are few so the increase is small.

### Less is more

The anti-intuitive fact that a thinner substrate gives less amplification of the pinhole diffusion is even more powerful when it comes to multi-layer barriers. If you have two thin "perfect" barrier materials with pinholes, separated by a layer of adhesive or polymer, the barrier properties decrease rapidly as the intermediate layer increases in thickness from, say, 0.1μm to 1μm - and then asymptotes. This means that super-planarised barrier coatings stuck together by super-thin, super-perfect intermediate layers can be orders of magnitude better than the same barrier materials coated "normally" with a comfortable few μm of intermediate coating. If the intermediate has some good barrier properties then another order of magnitude barrier can be obtained - though there is no letup on the need for it to be thin.

### Wasting your time and money

The Coverage and Pinholes models assume that your barrier is perfect other than the pinholes. It takes only a few pinholes to render the most amazing barrier rather useless. So before you waste your time and money on trying out the latest wonder barrier material, make sure you have the basics correct and eliminate as many pinholes as possible through attention to detail and to proper surface preparation (e.g. plasma treatments have been shown to make no difference to "surface energy adhesion" but they can greatly reduce pinholes). Even if you are pinhole-free, the "perfection" of a barrier material depends on lack of porosity in the barrier. Experience shows that if the first few layers of atoms aren't nicely compacted then adding more atoms on top is not as productive as it sounds (the atoms can't diffuse down to fill the holes). And if you add lots of atoms to really give a non-porous barrier it might be so thick that it now cracks when handled. Assuming that your process is such that it should give a good barrier and has low levels of pinholes then the following 3 steps are vital to avoid wasting even more time and money on a process that can't possibly deliver super barrier properties:

1. Eliminate all surface contamination so that compaction becomes possible. If you don't do this, then forget (2) and (3).
2. Minimise the porosity caused by nucleation and crystal growth issues, usually by the use of additional energy such as plasma densification during deposition
3. Only if (1) and (2) are optimised may it be worth considering a different material that has a better intrinsic barrier performance
1W Decker and B Henry, Basic Principles of Thin Film Barrier Coatings, Society of Vacuum Coaters 45th Annual Technical Conference Proceedings (2002) ISSN 0737-5921