0
1
2
3
0
1
2
3
h,  h
p
,   
φ 0.6
0.65 0.7 0.75 x/L
0
1
2
3
a
b
c
FIG. 7: (Colour online) A sequence of proﬁles during a dewetting process with competing evaporation and
convection that leads to the dried-in ring structure of nanoparticles displayed in Fig. 6(b). Proﬁles are at (a)
before pinning (t = 0.08T), (b) at self-pinning ( t = 0.13T), and (c) after depinning ( t = 0.29T), where
T = 3 ×1010τ with τ = η0γH/κ2 (T is of order of 1s). The ﬁlm thickness proﬁles hare the bold solid
lines, the nanoparticle concentrations φare the dotted lines and the nanoparticle layer height hp = hφare
the dashed lines. The remaining parameters and scalings are as in Fig. 6(b).
(i) in Fig. 8). The concentration increases further and when it approaches random close packing
φc, the viscosity diverges and the front pins itself. When pinned, further retraction only occurs
through evaporation (Fig. 7(b) and regime (ii) in Fig. 8). The front eventually depins and starts
to move again, leaving a nanoparticle ring behind (Fig. 7(c) and regime (iii) in Fig. 8). However,
the velocity is not as large as at the beginning, owing to the fact that the mean concentration of
particles has increased. The remaining particles are transported to the centre and are deposited
there when the remaining solvent evaporates (regime (iv) in Fig. 8).
The simple model used here shows, (i) that the contact line stops due to self-pinning by the de-
posited particles and (ii) the Marangoni effect is not necessary for the ring formation. The model
can easily be reﬁned to account for solutal and/or thermal Marangoni effects [88] but self-pinning
22