# A model for roughness statistics of heterogeneous fibrous materials

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## Abstract

We present a theory giving the standard roughness metrics \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) of stochastic fibrous materials in terms of their porosity and the thickness of the constituent fibres. Our treatment shows also that \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) are linearly dependent on each other with gradient depending on the skewness of the surface depth profile. Comparison of our theory with experiments on laboratory-formed paper samples and with data from the literature for industrially manufactured paper and carbon fibre nonwovens for use in fuel cells is excellent. The theory has applicability to generic families of stochastic fibrous materials and has relevance, for example, to printing of devices on paper.

## Background

There is a growing interest in the potential of paper as a flexible substrate for functional coatings, particularly of nanomaterials, and printed electronics. Recent examples include flexible chemiresistor sensors from coatings of carbon nanotubes [1] and graphene [2], flexible silver nanowire circuits, e.g. foldable RFID antennae [3], and optoelectronic devices from graphene on paper [4].

*d*is measured in a discrete process from a datum located at the highest point, yielding

*n*uniformly spaced measurements of \(d_i\). The roughnesses \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) are given by

*d*.

*t*. We note that the exponential distribution has standard deviation equal to the mean, so this dependency on fibre thickness will persist also in \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\). Niskanen et al. [25] used Poisson statistics to estimate \(R_{\mathrm{q}}\) for a random fibre network as

*A*is the area of a measuring head, such as a stylus or laser spot, and \(m_{\mathrm{f}}\) is the mass of a fibre. The dependence of \(R_{\mathrm{q}}\) on fibre mass predicted by Eq. (3) is in agreement with Dodson’s prediction of the effect of fibre thickness, since thicker fibres will be heavier than thinner ones of the same length; it is confirmed also by the experimental results of Li and Green [26]. However, the dependence of \(R_{\mathrm{q}}\) on network thickness and areal density provided by Eq. (3) is at variance with our intuitive understanding of roughness as a surface property. In part, the dependence can be attributed to network apparent density, \(\rho _{\mathrm{a}}\), which is given by the ratio \(\bar{\beta }/\bar{z}\) [27], such that

Here we derive expressions for the relationship between \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) for random fibre networks and provide theory giving these in terms of fibre dimensions and sheet properties. We compare our results with data from the literature and with those arising from experiments using papers with a range of densities formed in the laboratory using different fibres.

## Theory

*d*, from a datum located at the highest point in a given sample to the rough surface of a fibre network, as shown in Fig. 1. From our earlier discussion, we expect the distribution of

*d*to exhibit a positive skew. A good candidate distribution for,

*d*, is the gamma distribution, which has been widely applied to the modelling of pore size in a broad family of materials [29, 30], including stochastic fibrous materials [31, 32, 33]. The gamma distribution has probability density

*d*for random structures when \(k = 1\) and that of ‘near random’ structures, with variance dependent on parameter \(k > 0\) [34].

*in-plane*structure of paper is known to exhibit greater variability than a random network formed from the same constituent fibres due to flocculation in the forming process [23]; in contrast, we expect processes such as pressing and drying against smooth plates to remove small asperities from fibre surfaces giving a slightly lower variability in the measured distance

*d*and hence reduced roughness. As such, the range of application for our equations is \(k \ge 1\).

*within*a network with porosity \(\varepsilon \) formed from fibres with thickness

*t*is

## Experiment

Experiments were carried out to form paper samples in the laboratory according to international standards (TAPPI T-205 sp-02) and to measure their surface roughness. To test the dependence on fibre thickness, *t*, wood pulps with different morphologies were selected for papermaking: two hardwoods, birch (Södra Gold), eucalyptus (Portucel), and two softwoods, spruce (Mercer International) and pine (Stora Enso, Lapponia). All pulps were commercially produced by the Kraft process and bleached. To test the predicted dependency of roughness on sheet porosity, pulps were beaten in a Valley beater (TAPPI T-200 sp-01). Beaten pulps were sampled at 30-min intervals and 60 \(\mathrm {g}\,\mathrm{m}^{-2}\) handsheets made according to standard (TAPPI T-205 sp-02); note that the standard specifies that sheets are in contact with a polished steel plate on one side during the pressing and drying phases of manufacture, resulting in this side being smoother than the other.

Fibre dimensions for unbeaten fibres

Length | Width | Linear density | Thickness | |
---|---|---|---|---|

(mm) | (\(\upmu \mathrm {m}\)) | (\(\mathrm {\upmu g\,m^{-1}}\)) | (\(\upmu \mathrm {m}\)) | |

Pine | 2.0 | 20.3 | 132 | 4.2 |

Spruce | 2.3 | 21.1 | 146 | 4.4 |

Eucalyptus | 0.9 | 13.0 | 58 | 2.9 |

Birch | 0.8 | 14.1 | 68 | 3.1 |

## Results and discussion

*fibre*roughness to the sheet roughness on the plate-dried side than on the rough side.

*d*, whereas for \(k > 1\), the gamma distribution exhibits a maximum at \((k-1)\,\bar{d}/k\). Accordingly, we can interpret the higher value of

*k*obtained for the smooth side of our samples as a reduced contribution of fibre roughness to that of the sheet, such that the probability of small

*d*is less for the smooth side than for the rough, resulting in a maximum in the distribution of

*d*. For completeness, we note that the intercept of relevance is that when \(\varepsilon = 0\) such that \((1 + \varepsilon )/(1-\varepsilon ) = 1\). Given our assumptions and approximate method for determining porosity, it is perhaps unsurprising that this is nonzero. Nonetheless, it is encouraging that this offset is similar for both sides of our samples.

## Conclusions

We have presented a statistical model for the roughness of random and near random heterogeneous fibrous materials. Our theory uses the gamma distribution to represent the distribution of heights of a surface profile and predicts that the relationship between the standard roughness metrics \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) is approximately linear and bound within a narrow envelope. Comparison with data from our experiments on paper samples with a range of densities formed from different fibre types confirms this prediction, as do data from the literature for commercially produced papers and for carbon fibre nonwovens for fuel cells. Further, we have provided equations, showing that the absolute values of \(R_{\mathrm{a}}\) and \(R_{\mathrm{q}}\) are proportional to the thickness of constituent fibres and dependent on a simple function of network porosity. Agreement with data from our experimental paper samples is very good.

## Notes

### Acknowledgements

We gratefully acknowledge the work of Dr. Chris Wilkins in obtaining the micrographs in Figs. 3 and 4. We would like to thank Prof. Mike Turner for allowing us to use the Dektak profilometer and Dr. Adam Parry for assisting DW in its operation.

### Author Contributions

DW carried out the experimental work and developed the theory for the random case under the supervision of WWS. WWS extended the model to the general case and prepared the first draft of the manuscript; both authors contributed to the final version of the manuscript.

### Compliance with ethical standards

### Conflict of interest

The authors are unaware of any conflicts of interest regarding the content of this manuscript.

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