When soft matter solutions are dried, a skin layer is often observed at the surface of the solutions.^[1–3] This phenomenon is commonly observed in our daily life, such as cooling of hot milk and drying paint; it also appears in many industrial applications, such as polymer film production, ink-jet printing, etc. It has been reported that when the skin layer is formed, the system often shows various instabilities,^[4–9] including wrinkling, buckling and cavitation. Therefore understanding this process, especially when the skin layer forms and how it grows with time is an important issue.

Many theoretical studies have been done about the dynamics of the concentration profile of a solute (the non-volatile component) in drying solutions. The conventional approach is to solve the diffusion equation for solute concentration under moving boundary conditions due to the evaporation.^[10–12] This approach is useful when the system remains in a liquid phase, but it becomes difficult when some part of the system turns into a gel phase, where the evolution equation takes a different form.^[13,14] Recently, we have proposed a new scheme, called solute based Lagrangian scheme (Ref. [13]) to overcome the difficulty. This scheme utilizes the Lagrangian viewpoint of the solute motion, and describes the dynamics by tracing the average positions of the solutes. The scheme is suitable to deal with the moving boundaries and phase changes.

In this work, we will use this scheme to study the skin formation in a film of soft matter solutions placed on a substrate. We will numerically simulate the diffusion of the solutes, skin formation and its growth. We will also discuss how the elastic properties of the gel, the initial concentration of the solution, and the evaporation rate affect the skin formation.

This work is organized as follows. In Section 2, we will model the phenomenon of skin formation; in Section 3, we will describe the solute based Lagrangian scheme used in our calculation; in Section 4, we will numerically simulate the skin formation and its growth; and in Section 5, a brief conclusion is provided.

We consider a layer of soft matter solutions placed on a flat substrate whose normal direction lies in the x axis (Fig. 1).

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Evolution of the state of a soft matter solution film during drying. (a) the initial state, (b) the state of a non-uniform solution, (c) the state where the solution and the gel coexist, and (d) the final state of a uniform gel.

We assume that the volume fraction of the solute is initially uniform, denoted by ϕ₀ and the thickness of the layer is h₀ (Fig. 1(a)). With the evaporation of the solvents, the thickness h decreases and the volume fraction ϕ near the free surface increases (Fig. 1(b)). When the local volume fraction of the solutes ϕ reaches a specific value ϕ_g (gelation point), this part is treated as a gel (Fig. 1(c)). The gel layer grows with time and eventually occupies the whole system (Fig. 1(d)).

There are several approaches to obtaining a solute based Lagrangian scheme, and here we use the Onsager principle.^[14–16]

The Onsager principle determines the time evolution of slow variables which describe the non-equilibrium state of the system. Here we choose the positions of the solutes [X (x,t)] as the slow variable which denotes the average position of the solutes at time t that were located at x at time t = 0. According to the Onsager principle, Ẋ (x,t), the time derivative of X (x,t), is obtained by minimizing the following functional of X (x,t) and Ẋ (x,t), called Rayleighian:

Taking the functional variation of the Rayleighian with respect to Ẋ,

The evolution equation in the gel phase can be obtained in the same way. Solutions and gels are different in the free energy density form. The free energy density of a solution can be expressed as a function of the volume fraction ϕ(X;t), but the free energy of a gel, in general, cannot be represented by ϕ(X;t) alone: we need another state variable, the strain tensor, to express the free energy density of a gel. This can be seen in our previous work.^[13] However, in the present problem of the drying of a thin film, the strain can be represented by volume fraction, and the free energy density can be expressed by ϕ(X;t) alone. Here we assume that the energy density of a gel is written as

In our numerical calculation, we take h₀, K_BT/v_c and ζ as the basic units of the system. Diffusion constant D₀ = K_BT/v_cζ then becomes 1. Similarly, the reduced time becomes t_r = t/t₀ with t₀ equal to , and the reduced evaporation rate becomes J_r = J/J₀ with J₀ equal to D₀/h₀.

4.1. Validation of solute based Lagrangian scheme in simple solution

We first conduct a calculation to test the solutes based Lagrangian scheme, by comparing the numerical result of this method with that of the conventional one for a simple situation where there is no gelation. In this situation, the elastic parameter ε is zero, and the diffusion equation is written as

The boundary conditions are

The results of these two methods are compared in Fig. 2. It is seen that both approaches give precisely the same results.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Comparison between the solute based Lagrangian scheme (solid line) and the conventional method by solving the diffusion equation (cross line). The initial state ϕ0 = 0.1, and the evaporation speed J/J0 = 10.

4.2. Skin formation

Figure 3 shows the results of the numerical solution for Eq. (11) for varying the elastic parameter ε. Here the volume fraction ϕ_g is taken to be 0.2, and the dynamics is switched from the sol equation (15) to the gel equation (12) when ϕ exceeds 0.2. It is seen that for small ε the change from that of a pure solution is not so obvious, but for large ε the change is obvious: the concentration in the gel phase becomes almost a constant, and the slope of the curve is discontinuous at the sol/gel interface. Such volume fraction profiles were proposed in Ref. [11], but have not been confirmed. From Fig. 3, one can tell that gelation quickens the diffusion of solute apart from the evaporation surface.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Volume fraction profile for different elastic constant ε. (a) ε = 0, (b) ε = 1, (c) ε = 20, and (d) ε = 100. The other parameters are ϕ0 = 0.1, Jr = 10, ϕg = 0.2.

To discuss when a clear skin phase appears, we fix ε at 100, and solve the evolution equation by changing the evaporation rate and the initial volume fraction. The results are shown in Fig. 4. It can be known that the skin forms when both initial volume fraction and evaporation are large. In Figs. 4(b) and 4(c), a clear skin phase forms, but in Fig. 4(a) it does not. In Fig. 4(b) a clear skin phase forms, but in Fig. 4(d) it does not. When the initial concentration is high, the solution can become a gel early, which provides a positive feedback to the growth of the gel layer, due to a quicker diffusion of the solute towards the substrate surface. When the evaporation rate is large, the solution at the surface becomes a gel quicker, and there is a clear interface between the solution and the gel. Once the gel layer forms, it will grow quickly due to a promoted diffusion of the solutes.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Evolution of the volume fraction profile for various initial volume fraction ϕ0 and evaporation rate J/J0. The other parameters are ε = 100 and ϕg = 0.2. Comparison between panels (a) and (b) indicates that the initial volume fraction ϕ0 affects the skin formation, while a comparison between panels (b) and (d) indicates that e vaporation speed J/J0 is also important in the skin formation.

To define when the skin layer is formed, we introduce a parameter

where t_g is the time at which the gelation starts at the surface of the solution, and ϕ(0, t_g) is the volume fraction at the bottom of the solution when this happens. The parameter G stands for the comparison between the surface and the bottom when the gelation starts. If G is small, the skin phase is distinctively different from the solution phase, and we regard such a case as the case that a skin phase is formed. In Fig. 5, we have plotted G as a function of evaporation rate and the initial volume fraction. The figure clearly shows that the skin phase appears when both the initial volume fraction and the evaporation rate are large. This is in agreement with the observation of Ref. [11]. Figure 5 also includes the Peclet number line which was proposed in Ref. [11] to represent the condition for the appearance of the skin phase. The line is in agreement with the contour of G at G = 0.3, and therefore is consistent with the criterion we propose in this paper.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Gelation number G = (ϕ (0, tg) − ϕ0)/(ϕg − ϕ0) and Pe line Pe > (ϕg − ϕ0)/(1 − ϕ0)ϕ0.

4.3. Growth of skin phase

At the beginning of the drying process, the volume fraction of the solutes at the free surface increases, due to evaporation. When ϕ reaches a critical value ϕ_g, the soft matter solution becomes a gel. As the evaporation goes on, the thickness of the gel becomes larger.

Figure 6(a) shows an example of how skin grows. It is seen that after the free surface becomes a gel, the skin grows with time. However, the skin starts to shrink when the whole solution becomes a gel.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. Growth of the skin layer. Parameters are J/J0 = 10, ϕ0 = 0.1, ϕ0, ε = 100.

Figure 6(b) shows the growth rate of the gel as a function of the evaporation rate. It is seen that the growth rate is in proportion to evaporation rate J. This can be understood as follows. Consider the situation at a large evaporation rate. In this case, when the gel forms, the volume fraction at the bottom remains at ϕ₀, while the volume fraction at the surface remains at ϕ_g. Then the conservation of solute and solvent indicate that JΔt + Δh_g is equal to Δh_gϕ_g/ϕ₀. Hence the growth rate of the gel phase, Δh_g/Δ_t, is obtained as J(ϕ_g/ϕ₀ − 1). This expression is compared with that obtained by numerical calculation in Fig. 6(b). The agreement is good.

We have proposed a new model which describes the drying behavior of a thin film of soft matter solutions placed on a flat substrate. We have conducted extensive numerical calculations to study the dynamics of the skin phase formation, especially how the dynamics depend on the elastic parameter, the evaporation rate and the initial volume fraction. We have discussed the growth rate of the skin layer, which is shown to be linear with the evaporation rate.

The realistic case can be more complex than our simple model. Evaporation rate can change with time due to the evaporation, rather than keeping as a constant. Moreover, heat transfer can influence the diffusion process, while we neglect its effects for simplicity in this work. In spite of these, we believe that this work can provide a basic tool in handling the film formation process of drying soft matter solutions on a flat substrate.