2.1. Set-up

The experimental set-up is a ‘simplified 3-D printer’ (see figure 1). The fluid was injected, using a syringe pump, at a constant rate $Q$ through a hydrophobic nozzle with inner diameter 0.912 mm. The substrate was a glass plate, pre-wetted with a thin layer of the same fluid at constant height $h_{\infty } \approx 40\ \mathrm {\mu }{\rm m}$. The pre-wetting layer not only simulates the condition of a fully wetted surface but also allows us to avoid any complexity due to the (triple) contact line (Oron, Davis & Bankoff Reference Oron, Davis and Bankoff1997; Craster & Matar Reference Craster and Matar2009; Jalaal et al. Reference Jalaal, Stoeber and Balmforth2021; Martouzet et al. Reference Martouzet, Jørgensen, Pelet, Biance and Barentin2021). Pre-wetting was done just before the experiments to minimize evaporation effects. The distance between the nozzle and the surface of the pre-wetted film was fixed at $350\pm 10\ \mathrm {\mu }{\rm m}$. Changing this distance by $75\unicode{x2013}400\ \mathrm {\mu }{\rm m}$ did not change the results. The substrate was placed on a linear motor (Thorlabs LTS300/M). When the flow is established through the nozzle, the linear motor moves (constant velocity $U=10\ {\rm mm}\ {\rm s}^{-1}$ and initial acceleration $50\ {\rm mm}\ {\rm s}^{-2}$) and a viscoplastic line begins to form on the surface. The substrate motion continued until a line of length $80\pm 2\ {\rm mm}$ was obtained. Note that the short delays between the pump's function and the substrate's motion disturb the filament's shape at the beginning and end (${\sim }5\ {\rm mm}$ on each side). We perform the measurements far from these areas. Also, measurements were performed 20 s after the deposition to ensure that the final equilibrium shape was obtained. As the line is printed, it travels under a lens of an optical coherence tomography (OCT) scanner to obtain the shape of the interface. OCT is a non-invasive optical technique that uses low-coherence interferometry to obtain depth scans (Fercher et al. Reference Fercher, Drexler, Hitzenberger and Lasser2003). A series of these depth scans is then used to form a cross-sectional image. The technique has been used to study evaporating droplets (Edwards et al. Reference Edwards, Atkinson, Cheung, Liang, Fairhurst and Ouali2018), flow inside channels (Daneshi et al. Reference Daneshi, Pourzahedi, Martinez and Grecov2019), and spreading and solidification of droplets of complex fluids (Jalaal et al. Reference Jalaal, Seyfert, Stoeber and Balmforth2018). The current experimental set-up did not allow for an accurate measurement of dynamic of spreading. Nonetheless, using OCT provides many advantages for our system. In the set-up, the flat pre-wetted film, unlike the curved surface of the printed line, forms high-intensity patterns of vertical lines. This is due to the over-saturation of the reflected light on a flat surface. Hence, by detecting the edge of these patterns (vertical dashed white lines in figure 1), we were able to obtain easily the width of the filament with accuracy ${O}(10\ \mathrm {\mu }{\rm m}$). Furthermore, by adding microparticles ($3\ \mathrm {\mu }{\rm m}$ polystyrene from microparticles GmbH) to the fluid, we were able to roughly inspect the flow inside the printed lines and ensure that the flow along the length of the filament ($z$-direction in figure 1) was negligible, and therefore the spreading was mainly across the width ($x$–$y$ plane in figure 1). Experiments were repeated at least 10 times for each data point (total of $\sim$300 experiments). All measurements were performed at steady state when the filaments had reached their final shape. For each experiment, the OCT scans were performed for four cross-sections of the line, and the values were averaged.

Figure 1.

Experimental set-up for the investigation of printing a single line of viscoplastic fluids. The shown dimensions in the schematic are not to scale. A filament of a yield stress (viscoplastic) fluid is printed on a moving pre-wetted substrate. After extrusion from the nozzle, the filament spreads on the surface until it reaches a final width. We denote the final half-width of the filament as $\mathcal {R}_f$. An optical coherence tomography (OCT) scanner is used to measure the final width of the filaments. An example of the OCT image is shown in the top-left of this figure, where blue and red lines highlight the interface of the filament and the substrate. The dashed vertical lines show the edge of the filament, where the surface becomes flat.

2.2. Fluids

We used a mixture of Milli-Q water and a commercial hair gel at five different concentrations. The final materials are basically aqueous mixtures of Carbopol that are pH-neutralized with triethanolamine (Dinkgreve et al. Reference Dinkgreve, Paredes, Denn and Bonn2016). The samples were centrifuged before the experiments to remove bubbles. An Anton Paar MCR 302 rheometer with a rough cone-and-plate configuration ($2^{\circ }$ angle) was used to characterize the rheological properties of the liquids. For each sample, we obtained the steady-state flow curves as well as viscoelastic response in an oscillatory shear test. Figure 2 shows an example of measurements for one of the samples. Similar to what has been reported previously (Kim et al. Reference Kim, Song, Lee and Park2003; Dinkgreve et al. Reference Dinkgreve, Paredes, Denn and Bonn2016; Jalaal et al. Reference Jalaal, Schaarsberg, Visser and Lohse2019b), the fluids show a Herschel–Bulkley-type behaviour where the viscosities drop with the shear rate, and at low shear rates, the stress values converge to the yield stress. We therefore characterize the material properties by fitting the following equations to the flow curves (see figure 2a):

(2.1)\begin{equation} \tau - \tau_0 = K \dot{\gamma}^n, \end{equation}

where $\tau _0$ is the yield stress, $K$ is the consistency index, and $n$ is the shear index. The fitting values for six samples are listed in table 1. Oscillatory tests reveal the elastic-dominated behaviour at low strains/stresses with almost constant elastic and loss moduli. We quantified these values by averaging the values of $G^{\prime }$ and $G^{\prime \prime }$ at a small strain region, when $\gamma <1$ (see table 1). At large deformations, when stress exceeds the yield stress, the values of $G^{\prime }$ and $G^{\prime \prime }$ drop significantly, and the material behaves mostly like a viscous fluid with weak elasticity (see also Donley et al. Reference Donley, Singh, Shetty and Rogers2020).

Figure 2.

(a) Flow curve (shear stress versus shear rate) for sample 4 (see table 1). The inset shows the viscous stress ($\tau -\tau _0$) versus the shear rate. The symbols are the measurements, and the red lines are the corresponding Herschel–Bulkley fits. (b) Storage modulus ($G^{\prime }$, circles) and loss modulus ($G^{\prime \prime }$, squares) for sample 3 as a function of strain. The vertical grey line shows the critical strain at which stress is equal to the yield stress (from the Herschel–Bulkley fit). The inset shows the variation of shear stress versus the strain. The red line is a linear fit with the slope $G^{\prime }_0 \approx 223$. The normal lines highlight the critical strain and the yield stress.

Table 1.

Values of yield stress $\tau _0$, consistency index $K$, shear index $n$, small strain elastic modulus $G^{\prime }_0$ and loss modulus $G^{\prime \prime }_0$.

The densities of the fluids ($\rho$) are almost the same as for water (Jalaal et al. Reference Jalaal, Schaarsberg, Visser and Lohse2019b). Like other yield stress fluids, the surface tension of the materials ($\sigma$) is hard to measure. Previous attempts in the literature (Jørgensen et al. Reference Jørgensen, Le Merrer, Delanoë-Ayari and Barentin2015) mostly report slightly smaller values than the surface tension of water. We therefore use $\sigma = 0.072\ {\rm N}\ {\rm m}^{-1}$ in our calculations, but one should take this value likely as an upper bound.

3.1. Scaling law

Consider a viscoplastic filament that is spreading under capillary actions and comes to a halt due to the yield stress. The final shape of such a filament could be approximated by balancing the forces acting on the filament (Jalaal et al. Reference Jalaal, Stoeber and Balmforth2021). In this case, the driving force ($F_{\sigma }$) is due to the capillary action, and the resisting force is due to the yield stress ($F_{\tau _0}$), acting mainly on the base of the filament. These forces can be estimated as

(3.2a,b)\begin{equation} F_{\sigma} = \sigma \kappa \mathcal{H}_f b \quad \mathrm{and} \quad F_{\tau_0} = \tau_0 \mathcal{R}_f b, \end{equation}

where $\mathcal {H}_f$ is the final height of the filament, $\kappa \approx \mathcal {H}_f/\mathcal {R}_f^2$ is a characteristic curvature, and $b$ is the length of the filament. The cross-sectional area of the filament could be estimated as $A =\mathcal {L}^2\sim \mathcal {H}_f\mathcal {R}_f$. Using this, and balancing the forces in (3.2a,b), we arrive at

(3.3)\begin{equation} \mathcal{R}_f / \mathcal{L} = \varOmega \mathcal{J}^{{-}1/5}, \end{equation}

where $\varOmega$ is a prefactor. Equation (3.3) reveals a relatively weak power-law relationship between the normalized half-width of the filament and the plastocapillary number; as the plastic effects become larger, the filament spreads less. Generally, the values of prefactor $\varOmega$ can be a function of gravity, pre-wetted film thickness, elasticity, etc. However, for the range of experimental parameters presented here, $\varOmega$ can be estimated as a constant, representing the limit of pure plastocapillarity. In the following, we present a thin-film approximation of the present problem, in which we also aim to find the value of $\varOmega$.

3.2. Thin-film approximation

We apply a lubrication theory for a shallow and inertialess flow of Bingham viscoplastic fluids (shear index $n=1$) without gravity effects. Following the previous analyses of viscoplastic lubrication theory (Balmforth Reference Balmforth2019), and after scaling the length by $\mathcal {L}$, velocity by $\sigma /\mu$, and time by $\mathcal {L}\mu /\sigma$, we arrive at the following dimensionless depth-integrated evolution equation:

(3.4)\begin{equation} h_t= \frac{1}{6} \left[- h_{xxx} Y^2 \left(3h - Y \right) \right]_x, \quad \mathrm{with}\ Y = \mathrm{max}\left(0, h - \mathcal{J} / | h_{xxx}|\right), \end{equation}

where $h(x,t)$ is the height of the filament, and subscripts denote partial derivatives. In (3.4), $Y(x,t)$ is the yield surface, corresponding to the position in which the stress is equal to the yield stress. When $\mathcal {J} = 0$, we have $Y=h$ and hence (3.4) reduces to the well-known thin-film equation for Newtonian fluids (Oron et al. Reference Oron, Davis and Bankoff1997; Craster & Matar Reference Craster and Matar2009). Equation (3.4) represents the planar (two-dimensional) version of the viscoplastic lubrication theory for droplet spreading (Jalaal et al. Reference Jalaal, Stoeber and Balmforth2021). Note that the dynamics of the thin film spreading likely differs for Herschel–Bulkley fluids where $n\neq 1$; however, the final shape should still be well approximated in the limit studied here.

To explore the dynamics of filament spreading, we solve (3.4) using a finite difference method. In our solution, the length of the domain is large enough such that the filament never arrives at the boundaries. The initial shape of the filament was $h(x,t=0) = \max (0,1-x^2)^3 + h_{\infty }$ (Jalaal et al. Reference Jalaal, Stoeber and Balmforth2021). The first and second derivatives of the initial condition are zero at $x=1$ where it connects the interface of the filament to a pre-wetted film of thickness $h_{\infty }$ and has a height of almost unity at the centre with zero derivatives. Note that the pre-wetted film removes a genuine triple contact line and the complexities due to stress singularity (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009; Snoeijer & Andreotti Reference Snoeijer and Andreotti2013). Figure 4(a) shows the numerical solutions of the thin-film equation for $\mathcal {J}=0.02$. The filament yields fully under surface tension forces at $t=0$, and spreads on the surface. Later in the process, the influence of yield stress becomes more evident as the surface tension forces decrease. The filament eventually stops deforming when stress inside the filament falls below the yield stress ($Y \rightarrow 0$). The variations of half-width $\mathcal {R}(t)$ and maximum height $\mathcal {H}(t)$ of spreading filaments for different plastocapillary numbers are shown in figure 4(d,e), respectively. In the beginning of the spreading, the dynamics follows the Newtonian spreading (Tanner Reference Tanner1979), for which $\mathcal {R}(t) \sim t^{1/7}$ and $\mathcal {H}(t) \sim t^{-1/7}$, but eventually the curves deviate from the Newtonian limit and approach a constant value that corresponds to the final half-width and height of the filaments. The larger the value of the plastocapillary number, the sooner the filaments approach a final shape. Also, similar to what has been presented in the experiments, the filaments spread less as the value of the plastocapillary number increases.

Figure 4.

(a) Finite difference solution of (3.4). The 3-D dimensional plot is a quasi-representation of a line printing with $\mathcal {J}=0.02$ at $t=10^4$. The solid black line notes the final half-width. (b) The cross-section of the line at five different times. (c) Comparison of the theoretical final shape (where $\hat {h}=h/\mathcal {H}_f$ and $\hat {x}=x/\mathcal {R}_f$) with two experimental examples of samples 1 and 6 at $Q=10\ {\rm ml}\ {\rm h}^{-1}$, where the width and height are normalized with the theoretical prediction. The variations of (d) half-width ($\mathcal {R}$) and (e) maximum height ($\mathcal {H}$) of the filament are shown as functions of time for different values of $\mathcal {J}$. The dashed black lines show the scaling laws for Newtonian ($\mathcal {J}=0$) limits.

To compare the theoretical results of the final shapes with the experiments, we use the asymptotic limit of (3.4) for $Y \rightarrow 0$, at which the flow is ceased. Hence in dimensional form, we have $\sigma h\,h_{xxx} = \tau _0$. It is more convenient to re-scale the equation at this limit with final height and half-width, i.e. $\hat {h}=h/\mathcal {H}_f$ and $\hat {x}=x/\mathcal {R}_f$. Hence

(3.5)\begin{equation} \hat{h} \, \hat{h}_{\hat{x} \hat{x} \hat{x}} = \varLambda, \quad \mathrm{where} \ \varLambda = \tau_0\mathcal{R}_f^3 / \sigma \mathcal{H}_f^2. \end{equation}

The equation above governs the final shape of the filament with constant cross-sectional area $A = 2 \mathcal {H}_f \mathcal {R}_f \mathcal {I}$, where $\mathcal {I} = \int _0^1 \hat {h} \,\mathrm {d}\kern0.06em \hat {x}$ is a priori unknown. From the definitions of $\varLambda$ and $A$, we find the same relationship as in (3.3), with $\varOmega = (\varLambda / \mathcal {I}^2)^{1/5}$. We solve (3.5) as a boundary value problem (cf. the recent work on droplet spreading for the details of numerical method and boundary conditions, Jalaal et al. Reference Jalaal, Stoeber and Balmforth2021) to find the final shape of the filament and $\varLambda \approx 3.53$ and $\mathcal {I} \approx 0.58$, and therefore $\varOmega \approx 1.59$. Figure 4(c) compares the obtained theoretical shape with two examples of experimental data with smallest and largest yield stress. While theory predicts the shape of the fluid with small yield stresses very well (red line versus the OCT image), such agreement is less pronounced for the fluid with a higher yield stress. A comparison of (3.3) with the obtained $\varOmega$ from the asymptotic analysis above is shown in figure 3(b). While the theoretical prediction works fine for small values of $\mathcal {J}$, the comparison suffers from discrepancies at higher values. Fitting the data with a power-law function results in $\mathcal {R}_f / \mathcal {L} = 1.859 \mathcal {J}^{-0.156}$, showing the theory underpredict the prefactor and overpredict the magnitude of the power law by about 15 % and 28 %, respectively. The difference between theory and experiments could be due to both theoretical and experimental limitations. We will discuss these limitations in the next section and provide suggestions for future work.