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Mechanosorptive creep in nanocellulose materials

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Abstract

The creep behavior of nanocellulose films and aerogels are studied in a dynamic moisture environment, which is crucial to their performance in packaging applications. For these materials, the creep rate under cyclic humidity conditions exceeds any constant humidity creep rate within the cycling range, a phenomenon known as mechanosorptive creep. By varying the sample thickness and relative humidity ramp rate, it is shown that mechanosorptive creep is not significantly affected by the through-thickness moisture gradient. It is also shown that cellulose nanofibril aerogels with high porosity display the same accelerated creep as films. Microstructures larger than the fibril diameter thus appear to be of secondary importance to mechanosorptive creep in nanocellulose materials, suggesting that the governing mechanism is found between molecular scales and the length-scales of the fibril diameter.

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Acknowledgements

Anne-Mari Olsson and Lennart Salmén, Innventia AB, are acknowledged for their assistance during the experiments. S. B. L. thanks BiMaC Innovation for financial support.

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Correspondence to Stefan B. Lindström.

Appendix 1: Fickian diffusion with exponential plateau boundary conditions

Appendix 1: Fickian diffusion with exponential plateau boundary conditions

Symmetric one-dimensional diffusion of moisture in a film with exponential plateau Dirichlet boundary conditions is solved in nondimensional form for brevity. Taking the initial state to be equilibrium and using the symmetry around the center of the film render

$$ u_{t} = u_{xx} $$
(8a)
$$ u_x(0,t) = 0 $$
(8b)
$$ u(1,t) = 1-e^{-\alpha t} $$
(8c)
$$ u(x,0) = 0, $$
(8d)

where \(x\in[0, 1]\) and α is a constant. Taking the Laplace transform gives

$$ sU - u(x,0) = U_{xx} $$
(9a)
$$ U_x(0,s) = 0 $$
(9b)
$$ U(1,s) = {\frac{1} {s}}-{\frac{1} {s + \alpha}}, $$
(9c)

where \({U(s)={\fancyscript L}[u(t)]}\). The solution of Eq. 9a is readily obtained as

$$ U(x,s) = A(s) e^{x\sqrt{s}} + B(s) e^{-x\sqrt{s}}. $$
(10)

Equation 10 and the boundary conditions (9b) and (9c) yield

$$ U(x,s) = {\frac{\cosh\sqrt{sx^2}} {\cosh\sqrt{s}}} \cdot {\frac{\alpha} {s(s + \alpha)}}. $$
(11)

Direct application of the Laplace transform inversion formula renders

$$ u(x,t) = \lim_{b\rightarrow\infty} {\frac{1} {2 \pi i}} \int\limits_{a-bi}^{a+bi} e^{st}U(x,s) {\rm d} s\,\quad a > 0. $$
(12)

Since the integrand is analytic in the complex plane, except at the poles s = 0, s =  −α and \(s = - \pi^2(n+1/2)^2, \quad n = 0,1,\ldots\), and since the integral vanishes when \(|s|\rightarrow \infty, \Re(s) < 0\), we may compute u(xt) as the sum of residues

$$ \begin{aligned} u(x,t) = \hbox{Res}[{e^{st}U(x,s)},{0}] + \hbox{Res}[{e^{st}U(x,s)},{-\alpha}] +\\ \sum_{n=0}^{\infty} \hbox{Res}[{e^{st}U(x,s)},{-\pi^2(n+1/2)^2}] \\= 1 - {\frac{\cos\sqrt{\alpha x^2}}{\cos\sqrt{\alpha}}} e^{-\alpha t} - 2 \sum_{n=0}^{\infty} (-1)^n {\frac{\alpha\cos(a_nx)} {a_n(\alpha-a_n^2)}} e^{-a_n^2 t}, \end{aligned} $$
(13)

where a n  = π(n + 1/2). Note that the equations in this appendix are all in nondimensional form.

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Lindström, S.B., Karabulut, E., Kulachenko, A. et al. Mechanosorptive creep in nanocellulose materials. Cellulose 19, 809–819 (2012). https://doi.org/10.1007/s10570-012-9665-9

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