Soft Matter View Article Online
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
PAPER
View Journal | View Issue
Dynamics of unidirectional drying of colloidal dispersions† Cite this: Soft Matter, 2014, 10, 4151
Pierre Lidon and Jean-Baptiste Salmon* We investigate the dynamics of unidirectional drying of silica dispersions. For small colloids (radii a < 15 nm), the minute recession of the drying interface inside the growing solid leads to a slowing down of the evaporation rate, as recently proposed by Wallenstein and Russel [J. Phys.: Condens. Matter, 2011, 23, 194104]. We first propose that Kelvin's effect, i.e. the reduction of the partial pressure of water in the presence of highly curved nanomenisci at the drying air–dispersion interface, has to be taken into account, notably for such small colloids. Our model can fit qualitatively the literature measurements, but with a crossover between the linear regime and the slowing down regime that scales as a2, as compared to the model of Wallenstein and Russel that predicts a linear scaling. We then also present careful
Received 27th September 2013 Accepted 11th March 2014
measurements of the dynamics of solidification, which clearly demonstrate that both models (taking into account or not Kelvin's effect) do not fit correctly the slowing down. This is consistent with a brief review of similar recent measurements. Nevertheless, the dynamics can be correctly estimated with a significantly lower effective permeability of the solid region. We suggest that this result may come from
DOI: 10.1039/c3sm52528g
the polydispersity of the suspensions, and from the inhomogeneity of the flow within the fracturated
www.rsc.org/softmatter
solid region, as illustrated by infiltration experiments of a coloured dye.
1
Introduction
Drying of colloidal dispersions is a common industrial process, as well as a fundamental step for engineering innovative materials.1 In this wide eld, physicists focussed their efforts to understand the mechanisms at work, thanks to various model experiments: drying of droplets,2,3 of lms,4,5 in microuidic channels.6,7Among the geometries investigated, conned unidirectional drying has received particularly a lot of attention since the pioneering studies of Allain et al.,8 and later of Dufresne et al.9,10 In these experiments, a small and long capillary (height 50–200 mm, width 0.1–2 mm, and length > 1 cm) is partially lled with an aqueous dispersion (oen charged-stabilized silica or polystyrene colloids with radii a in the 5–100 nm range), see Fig. 1. Evaporation of water only occurs from one tip of the capillary, as the water vapor rapidly saturates the other outlet. Evaporation induces a ow J towards the open end, which continuously convects the colloids at the air–dispersion interface. Colloids thus concentrate until the formation of a dense incompressible solid, which invades the capillary at a given pace y_ f. During the growth process, fascinating cracks and delamination occur within the solid region. As this model geometry offers a unique way to probe nely these complex
Univ. Bordeaux/CNRS/RHODIA, LOF, UMR 5258, 178, Avenue Schweitzer, F-33600 Pessac, France. E-mail: [email protected] † Electronic supplementary information (ESI) available: 3 movies and a pdf le. See DOI: 10.1039/c3sm52528g
This journal is © The Royal Society of Chemistry 2014
phenomena, many groups tried to provide a complete understanding of the underlying mechanisms using original experiments11–16 and dedicated models.17–20 Particularly, Dufresne et al. demonstrated, thanks to coherent anti-Stokes Raman scattering microscopy, that the
Fig. 1 Top: typical unidirectional drying experiment (perspective view). A capillary is partially filled with an aqueous dispersion and water evaporates at a rate J from one tip. Bottom: top view of the capillary. Colloids concentrate until the formation of a solid that invades the capillary. yf is the position of the compaction front, and ym is that of the air–dispersion meniscus. The solid region often displays a complex pattern of cracks (symbolized by parallel lines), and often delaminates from the capillary walls (the delamination front is symbolized by the dashed line).
Soft Matter, 2014, 10, 4151–4161 | 4151
View Article Online
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
Soft Matter
solid region remains wet during the growth (except for the cracks), and with a uniform water concentration.9,10 To explain this result, the same authors proposed that the strong wetting of the particles with water prevents the invasion of the colloidal solid by the drying air–water interface. Evaporation then induces a ow through the solid region, associated with a viscous dissipation, and with curved nanomenisci at the xed air–water interface. Moreover, two different regimes of growth have been clearly identied during the solidication process.9,13,14 (i) For large colloids (radii a > 20 nm) and/or for early times, the compact region invades the capillary at a constant velocity yf t, as expected from the solute conservation. (ii) In the case of smaller nanoparticles (and/or at longer time scales) the compaction front deviates from this linear regime, and follows yf t1/2, thus suggesting a slowing down of the evaporation rate J.9,13,14 To account for these observations, Dufresne et al. provided an empirical law describing qualitatively the experiments.9 However, this phenomenological law predicts that the crossover between the two regimes of growth occurs for a critical compaction front position y0 that scales linearly with a, in contradiction with their experimental data. An alternative theoretical picture was proposed later by Wallenstein and Russel to explain the slowing down20 (note that it was also proposed in a footnote of ref. 10). The slowing down of the growth dynamics is explained by a minute recession of the drying air–water interface inside the bulk of the colloidal packing for a critical position of the compaction front. This mechanism corresponds to a well-known observation of the drying of gels, porous media, and granular materials.17,18,21–24 For small colloids, the air invasion in the bulk of the solid region adds an additional transfer resistance for the removal of the vapor, and slows down the growth rate of the solid region. However, this model again predicts that the crossover between the two regimes occurs for a critical compaction front position y0 a, in contradiction with the experiments.9 Moreover, to our knowledge, no quantitative comparisons with experimental data have been provided to test this model. In the present paper, we rst show that the Kelvin effect, i.e. the decrease of the partial pressure of water (and thus of the evaporate rate) in the presence of highly curved menisci at the drying interface due to the high viscous dissipation along the dense colloidal packing, has to be taken into account in the model mentioned above,20 notably for small colloids (a < 15 nm). This modied model permits again to predict the two regimes (linear and square root growth), but with a slowing down that occurs sooner, at a crossover that scales as a2. We then performed experiments of unidirectional drying of silica nanoparticles, and we present measurements of both the compaction front yf(t) and the evaporation rate J(t). We show that both models (with or without Kelvin's effect) do not predict quantitatively the experiments when the permeability of the dense assembly is estimated using the classical Carman– Kozeny relationship. Such a result is consistent with a brief review of similar experiments found in the literature. We then propose possible explanations. We indeed performed preliminary experiments of inltration of a coloured dye through the
4152 | Soft Matter, 2014, 10, 4151–4161
Paper
solid region during drying. These experiments show that the ow within the colloidal solid is not uniform, and this may explain the discrepancy between our observations and the models. We also present TEM characterizations of the silica dispersions under study, and we suggest that the observed polydispersity and the deviation of the colloids from spherical nanoparticles may also explain our observations.
2 Growth dynamics of the solid region We rst detail briey the model of Wallenstein and Russel20 that describes the growth dynamics of the solid region. We then take into account the Kelvin effect that adds a new contribution for the slowing down of the evaporation rate, notably for small colloids (a < 15 nm). 2.1 Slowing down of the dynamics: receding of the drying interface In the experiment described in Fig. 1, drying is governed by two key factors only: (1) the difference between the chemical potential of water at the drying interface and the external humidity; and (2) the removal mechanism of the vapor from the drying interface (through vapor diffusion and convection in the room). In most of the experimental conditions investigated, induced-evaporation rates are of the order of J z 1 mm s1, and convect the colloids until the formation of a compact packing at the drying interface. This evaporation-induced ow is associated with a pressure gradient due to the viscous dissipation along the dense solid, and thus with curved nanomenisci at the air–dispersion interface to sustain the strong particle wetting.9 For a quasi-steady ow, this pressure drop is oen estimated, thanks to: dp ¼ hJyf/k,
(1)
where yf is the position of the compaction front, k the permeability of the compact packing, and h the viscosity of water (viscous dissipation is assumed to be negligible for y > yf in the liquid dispersion20). Solute conservation also implies: ðFd F0 Þ
dyf ¼ f0 J; dt
(2)
where Fd is the volume fraction of the compact packing, and F0 is that of the dilute dispersion (we comment on the validity of this equation in Section 3.2). For constant evaporation rates J ¼ J0 (and for a uniform compacity Fd), the growth velocity of the dense packing is constant, and yf follows simply: yf ¼
F0 J0 t; DF
(3)
with DF ¼ Fd F0. This linear growth yf t has been observed in the case of large colloids (a > 20 nm), or during the early stage in the case of smaller nanoparticles.9 The square root growth yf t1/2 observed for smaller colloids (and/or for longer time scales)9,13,14 is attributed to the increase of the pressure drop dp that reaches pm, corresponding to the
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
Paper
Soft Matter
maximal curvature of the nanomenisci before the receding of the interface in the bulk of the solid.20 Estimates of pm differ in the literature, and we will adopt in the following pm ¼ 2g/(0.15a) as done in ref. 9 and 10 for a better comparison with their experiments (g ¼ 70 mN m1 is the air–water surface tension). Numerical values will also be provided in the present work, thanks to the formula pm z 5.3g/a used for instance in ref. 17 and 20. This second regime occurs at a time t0 and for a compaction front y0 given by: y0 ¼ kpm/(hJ0), t0 ¼ (DF/F0)(y0/J0).
(4)
As soon as the drying interface recedes in the solid region, there is an additional transfer resistance for the removal of the vapor through the pores of the dry solid. The resistance of this mass transfer is oen assumed to be very large compared to the resistance of the transfer of the vapor from the tip of the capillary. Indeed, since the pore size is very small, the transport of vapor through the dry solid occurs by diffusion in the Knudsen regime.20 This explains that the receding of the interface is very small, and thus not observed in the experiments. In this regime, and as long as the resistance of the mass transfer through the dry solid remains very large, one has:20 Jyf z J0y0 ¼ kpm/h.
This p equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidisplays some similarities with the formula yf ¼ y0 ð 1 þ 2t=t0 1Þ; derived by Dufresne et al., thanks to an empirical combination of the evaporation rate J0 and the hydrodynamic ux J ¼ kVp/h.9 We now make three important remarks. First, as k a2 and pm a1, this model predicts y0 a. However, this linear scaling is in conict with the experiments performed using silica nanoparticles with different radii a (6–26 nm).9 Second, at long time scales (t [ t0), the compaction front follows yf2 z (2F0/DF)(kpm/h)t. The dynamics yf(t) should thus provide robust estimates of kpm/h (for a known Fd) at long time scales, since it does not depend on the value of J0. Finally, no clear quantitative comparisons have been done to our knowledge between eqn (3)–(6) and experimental data. Importantly, t0 and y0 are related to each other as y0/t0 ¼ J0F0/DF, and any comparison with the experiments should not take these two parameters as independent tting parameters. Modied model taking into account Kelvin's effect
During the drying process, the pressure drop dp due to the viscous dissipation in the solid region, continuously increases until the receding of the drying interface. At this stage, dp ¼ pm reaches very large values for small colloids as those investigated in ref. 9, 13 and 14 and in the present work (a z 6–13 nm): pm z This journal is © The Royal Society of Chemistry 2014
where pk is the Kelvin pressure given by: pk ¼ kBT/vm z 1380 bar,
(8)
at room temperature (vm is the molecular volume of water). For radii a ¼ 6–13 nm, one has indeed J/J0 z 0.3–0.6 (J/J0 z 0.6–0.8 for pm ¼ 5.3g/a). We thus modied the model described by Wallenstein and Russel20 to take into account this effect. Before air invasion (and thus for dp < pm), the compaction front yf and the evaporation rate J follow the relationships: DFy_ f ¼ F0J, J ¼ J0 exp(dp/pk), dp ¼ hJyf/k.
(5)
We add a more complete derivation of this relationship in the ESI.† The dynamics of the compaction front can then be estimated according to the conservation equation (eqn (2)) leading to the square root behavior:20 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yf ¼ y0 1 þ 2ðt t0 Þ=t0 : (6)
2.2
720–1550 bar for pm ¼ 2g/(0.15a) (pm z 280–620 bar for pm ¼ 5.3g/a). For such huge pressure drops, one cannot neglect the decrease of the chemical potential of water at the drying interface due to the Kelvin law.23 It indeed yields a decrease of the evaporation rate according to: dp J ¼ J0 exp ; (7) pk
The possibly high viscous dissipation in the solid region thus decreases J(t), and consequently slows down the growth rate y_ f. We introduce the natural length scale yk ¼ kpk/(hJ0), and get the unitless equations: d~ yf ~ ¼ J; dt~ J~ ¼ exp(J~y~f),
(9)
(10)
with ~yf ¼ yf/yk, ~t ¼ t/tk, and ~J ¼ J/J0, where tk ¼ (DF/F0)(yk/J0). For small pressure drops compared to pk, one can linearize the exponential term, and nd simply the dynamics of the compaction front: sffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2t yf z yk 1þ 1 ; (11) tk Jz
J0 J0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: 1 þ yf yk 1 þ 2t=tk
(12)
Interestingly, this equation has a similar form ffias the one pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi found empirically in ref. 9 (yf ¼ y0 ð 1 þ 2t=t0 1Þ; see previous section), but with the important difference that yk a2, as compared with y0 that scales linearly with a. This difference may explain the observations of ref. 9, i.e. measurements of y0 for several a, which deviate from a linear scaling. The regime described above fails as soon as the pressure drop dp reaches its maximal value pm corresponding to the receding of the air–water interface. Again, there is an additional mass transfer resistance, and following the arguments provided by Wallenstein and Russel:20 Jyf z Jiyi ¼ kpm/h, where Ji < J0 is
Soft Matter, 2014, 10, 4151–4161 | 4153
View Article Online
Soft Matter
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
the evaporation rate when dp ¼ pm, and yi the corresponding position of the compaction front. At longer time scales, the dynamics follows again: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F0 Ji yf ¼ yi 1 þ 2 ðt ti Þ; (13) DFyi with ti given by yf(ti) ¼ yi. In the following, we use numerical solutions of eqn 9 and 10 to compute yf(t) and J(t) before the receding of the interface. Fig. 2 displays the dynamics of the compaction front and of the evaporation rate for a ¼ 13 nm and a ¼ 6 nm (pm ¼ 2g/ (0.15a)), for both models, i.e. with and without Kelvin's effect. At long time scales (t [ tk), both models lead to close dynamics for yf(t) and J(t). In this regime indeed, evaporation is mainly governed by the highly resistive mass transfer of vapor through the pores of the dry solid. At an early stage however, a clear difference is observed, mainly for the temporal behavior of J(t) (the difference between the two dynamics of yf(t) is less pronounced). Particularly, no constant J(t) is observed when taking into account Kelvin's law, and the receding of the drying interface occurs later due the initial decrease of J(t) because of the increase of the viscous dissipation. These behaviors depend strongly on the size of the colloids (as well as the numerical estimates of pm), and the differences between the two dynamics are more important in the case of small colloids, see the insets of Fig. 2 for a ¼ 6 nm. For a >
Paper
15 nm, the two different compaction front curves are almost indistinguishable, but a signicant decrease of J can always be observed at small time scales when taking into account the Kelvin effect (data not shown). In the following, we try to compare these predictions (along with the model of ref. 20) with experimental data, and experimental values inferred from recent studies.9,10,13,14
3 Comparison with experimental data 3.1
Measurements from the literature
Surprisingly, no clear attempts have been made to our knowledge to compare quantitatively experimental data to the theoretical predictions given above. The square root dynamics pffiffi yf t has been clearly identied in ref. 9, 10, 13 and 14, howeverpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi only ffi ts based on the empirical law yf ¼ y0 ð 1 þ 2t=t0 1Þ are provided, and more importantly with no clear demonstration that both y0 and t0 are not two independent parameters. Such a quantitative comparison should a priori lead to numerical values of the permeability k of the solid. Indeed, at long time scales, i.e. for a receding drying interface, measurements of both J(t) and yf(t) should give precise estimates of kpm as Jyf z kpm/h; the same measurements at an early stage may discriminate the two models given above (Fig. 2). Models and experiments that try to understand the formation of fractures and delamination during unidirectional drying use the Carman–Kozeny relationship: k ¼ a2
ð1 Fd Þ3 ; 45Fd 2
(14)
to compute the permeability k of the solid region. The latter is known to describe very well the ow through randomly closepacked monodisperse spheres,25 but despite an extensive literature survey, we do not nd any conrmation of the validity of this law for the small colloids typically investigated in the present work and in ref. 9, 10, 13 and 14 (silica particles, a < 15 nm). Such direct measurements are indeed probably difficult to perform due to the very high pressure drops involved, and the possible formation of fractures that can lead to experimental artifacts. Table 1 Experimental parameters corresponding to unidirectional drying experiments reported in ref. 10, 13 and 14 indicate values that are not measured (or not reported in the corresponding reference)
Fig. 2 Top: (a ¼ 13 nm) compaction front yf normalized by yk; model of ref. 20, and model taking into account Kelvin's effect. Both symbols indicate the transition to the regime of the receding of the drying interface. The dotted line indicates the linear growth given by eqn (3). Bottom: corresponding normalized evaporation rates J/J0. The two insets show the same plots but for a ¼ 6 nm.
4154 | Soft Matter, 2014, 10, 4151–4161
Ref.
Fig. 2 of ref. 10
Fig. 3 of ref. 13
Fig. 4 of ref. 14
Radius a F0 Fd J0 (mm s1) Cell (mm mm) y0 (exp.) y0, eqn (4) t0 (exp.) t0, eqn (4)
11 nm 0.2 0.58 1a 0.1 2 1 mm 5 (2) cm
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
PAPER
View Journal | View Issue
Dynamics of unidirectional drying of colloidal dispersions† Cite this: Soft Matter, 2014, 10, 4151
Pierre Lidon and Jean-Baptiste Salmon* We investigate the dynamics of unidirectional drying of silica dispersions. For small colloids (radii a < 15 nm), the minute recession of the drying interface inside the growing solid leads to a slowing down of the evaporation rate, as recently proposed by Wallenstein and Russel [J. Phys.: Condens. Matter, 2011, 23, 194104]. We first propose that Kelvin's effect, i.e. the reduction of the partial pressure of water in the presence of highly curved nanomenisci at the drying air–dispersion interface, has to be taken into account, notably for such small colloids. Our model can fit qualitatively the literature measurements, but with a crossover between the linear regime and the slowing down regime that scales as a2, as compared to the model of Wallenstein and Russel that predicts a linear scaling. We then also present careful
Received 27th September 2013 Accepted 11th March 2014
measurements of the dynamics of solidification, which clearly demonstrate that both models (taking into account or not Kelvin's effect) do not fit correctly the slowing down. This is consistent with a brief review of similar recent measurements. Nevertheless, the dynamics can be correctly estimated with a significantly lower effective permeability of the solid region. We suggest that this result may come from
DOI: 10.1039/c3sm52528g
the polydispersity of the suspensions, and from the inhomogeneity of the flow within the fracturated
www.rsc.org/softmatter
solid region, as illustrated by infiltration experiments of a coloured dye.
1
Introduction
Drying of colloidal dispersions is a common industrial process, as well as a fundamental step for engineering innovative materials.1 In this wide eld, physicists focussed their efforts to understand the mechanisms at work, thanks to various model experiments: drying of droplets,2,3 of lms,4,5 in microuidic channels.6,7Among the geometries investigated, conned unidirectional drying has received particularly a lot of attention since the pioneering studies of Allain et al.,8 and later of Dufresne et al.9,10 In these experiments, a small and long capillary (height 50–200 mm, width 0.1–2 mm, and length > 1 cm) is partially lled with an aqueous dispersion (oen charged-stabilized silica or polystyrene colloids with radii a in the 5–100 nm range), see Fig. 1. Evaporation of water only occurs from one tip of the capillary, as the water vapor rapidly saturates the other outlet. Evaporation induces a ow J towards the open end, which continuously convects the colloids at the air–dispersion interface. Colloids thus concentrate until the formation of a dense incompressible solid, which invades the capillary at a given pace y_ f. During the growth process, fascinating cracks and delamination occur within the solid region. As this model geometry offers a unique way to probe nely these complex
Univ. Bordeaux/CNRS/RHODIA, LOF, UMR 5258, 178, Avenue Schweitzer, F-33600 Pessac, France. E-mail: [email protected] † Electronic supplementary information (ESI) available: 3 movies and a pdf le. See DOI: 10.1039/c3sm52528g
This journal is © The Royal Society of Chemistry 2014
phenomena, many groups tried to provide a complete understanding of the underlying mechanisms using original experiments11–16 and dedicated models.17–20 Particularly, Dufresne et al. demonstrated, thanks to coherent anti-Stokes Raman scattering microscopy, that the
Fig. 1 Top: typical unidirectional drying experiment (perspective view). A capillary is partially filled with an aqueous dispersion and water evaporates at a rate J from one tip. Bottom: top view of the capillary. Colloids concentrate until the formation of a solid that invades the capillary. yf is the position of the compaction front, and ym is that of the air–dispersion meniscus. The solid region often displays a complex pattern of cracks (symbolized by parallel lines), and often delaminates from the capillary walls (the delamination front is symbolized by the dashed line).
Soft Matter, 2014, 10, 4151–4161 | 4151
View Article Online
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
Soft Matter
solid region remains wet during the growth (except for the cracks), and with a uniform water concentration.9,10 To explain this result, the same authors proposed that the strong wetting of the particles with water prevents the invasion of the colloidal solid by the drying air–water interface. Evaporation then induces a ow through the solid region, associated with a viscous dissipation, and with curved nanomenisci at the xed air–water interface. Moreover, two different regimes of growth have been clearly identied during the solidication process.9,13,14 (i) For large colloids (radii a > 20 nm) and/or for early times, the compact region invades the capillary at a constant velocity yf t, as expected from the solute conservation. (ii) In the case of smaller nanoparticles (and/or at longer time scales) the compaction front deviates from this linear regime, and follows yf t1/2, thus suggesting a slowing down of the evaporation rate J.9,13,14 To account for these observations, Dufresne et al. provided an empirical law describing qualitatively the experiments.9 However, this phenomenological law predicts that the crossover between the two regimes of growth occurs for a critical compaction front position y0 that scales linearly with a, in contradiction with their experimental data. An alternative theoretical picture was proposed later by Wallenstein and Russel to explain the slowing down20 (note that it was also proposed in a footnote of ref. 10). The slowing down of the growth dynamics is explained by a minute recession of the drying air–water interface inside the bulk of the colloidal packing for a critical position of the compaction front. This mechanism corresponds to a well-known observation of the drying of gels, porous media, and granular materials.17,18,21–24 For small colloids, the air invasion in the bulk of the solid region adds an additional transfer resistance for the removal of the vapor, and slows down the growth rate of the solid region. However, this model again predicts that the crossover between the two regimes occurs for a critical compaction front position y0 a, in contradiction with the experiments.9 Moreover, to our knowledge, no quantitative comparisons with experimental data have been provided to test this model. In the present paper, we rst show that the Kelvin effect, i.e. the decrease of the partial pressure of water (and thus of the evaporate rate) in the presence of highly curved menisci at the drying interface due to the high viscous dissipation along the dense colloidal packing, has to be taken into account in the model mentioned above,20 notably for small colloids (a < 15 nm). This modied model permits again to predict the two regimes (linear and square root growth), but with a slowing down that occurs sooner, at a crossover that scales as a2. We then performed experiments of unidirectional drying of silica nanoparticles, and we present measurements of both the compaction front yf(t) and the evaporation rate J(t). We show that both models (with or without Kelvin's effect) do not predict quantitatively the experiments when the permeability of the dense assembly is estimated using the classical Carman– Kozeny relationship. Such a result is consistent with a brief review of similar experiments found in the literature. We then propose possible explanations. We indeed performed preliminary experiments of inltration of a coloured dye through the
4152 | Soft Matter, 2014, 10, 4151–4161
Paper
solid region during drying. These experiments show that the ow within the colloidal solid is not uniform, and this may explain the discrepancy between our observations and the models. We also present TEM characterizations of the silica dispersions under study, and we suggest that the observed polydispersity and the deviation of the colloids from spherical nanoparticles may also explain our observations.
2 Growth dynamics of the solid region We rst detail briey the model of Wallenstein and Russel20 that describes the growth dynamics of the solid region. We then take into account the Kelvin effect that adds a new contribution for the slowing down of the evaporation rate, notably for small colloids (a < 15 nm). 2.1 Slowing down of the dynamics: receding of the drying interface In the experiment described in Fig. 1, drying is governed by two key factors only: (1) the difference between the chemical potential of water at the drying interface and the external humidity; and (2) the removal mechanism of the vapor from the drying interface (through vapor diffusion and convection in the room). In most of the experimental conditions investigated, induced-evaporation rates are of the order of J z 1 mm s1, and convect the colloids until the formation of a compact packing at the drying interface. This evaporation-induced ow is associated with a pressure gradient due to the viscous dissipation along the dense solid, and thus with curved nanomenisci at the air–dispersion interface to sustain the strong particle wetting.9 For a quasi-steady ow, this pressure drop is oen estimated, thanks to: dp ¼ hJyf/k,
(1)
where yf is the position of the compaction front, k the permeability of the compact packing, and h the viscosity of water (viscous dissipation is assumed to be negligible for y > yf in the liquid dispersion20). Solute conservation also implies: ðFd F0 Þ
dyf ¼ f0 J; dt
(2)
where Fd is the volume fraction of the compact packing, and F0 is that of the dilute dispersion (we comment on the validity of this equation in Section 3.2). For constant evaporation rates J ¼ J0 (and for a uniform compacity Fd), the growth velocity of the dense packing is constant, and yf follows simply: yf ¼
F0 J0 t; DF
(3)
with DF ¼ Fd F0. This linear growth yf t has been observed in the case of large colloids (a > 20 nm), or during the early stage in the case of smaller nanoparticles.9 The square root growth yf t1/2 observed for smaller colloids (and/or for longer time scales)9,13,14 is attributed to the increase of the pressure drop dp that reaches pm, corresponding to the
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
Paper
Soft Matter
maximal curvature of the nanomenisci before the receding of the interface in the bulk of the solid.20 Estimates of pm differ in the literature, and we will adopt in the following pm ¼ 2g/(0.15a) as done in ref. 9 and 10 for a better comparison with their experiments (g ¼ 70 mN m1 is the air–water surface tension). Numerical values will also be provided in the present work, thanks to the formula pm z 5.3g/a used for instance in ref. 17 and 20. This second regime occurs at a time t0 and for a compaction front y0 given by: y0 ¼ kpm/(hJ0), t0 ¼ (DF/F0)(y0/J0).
(4)
As soon as the drying interface recedes in the solid region, there is an additional transfer resistance for the removal of the vapor through the pores of the dry solid. The resistance of this mass transfer is oen assumed to be very large compared to the resistance of the transfer of the vapor from the tip of the capillary. Indeed, since the pore size is very small, the transport of vapor through the dry solid occurs by diffusion in the Knudsen regime.20 This explains that the receding of the interface is very small, and thus not observed in the experiments. In this regime, and as long as the resistance of the mass transfer through the dry solid remains very large, one has:20 Jyf z J0y0 ¼ kpm/h.
This p equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidisplays some similarities with the formula yf ¼ y0 ð 1 þ 2t=t0 1Þ; derived by Dufresne et al., thanks to an empirical combination of the evaporation rate J0 and the hydrodynamic ux J ¼ kVp/h.9 We now make three important remarks. First, as k a2 and pm a1, this model predicts y0 a. However, this linear scaling is in conict with the experiments performed using silica nanoparticles with different radii a (6–26 nm).9 Second, at long time scales (t [ t0), the compaction front follows yf2 z (2F0/DF)(kpm/h)t. The dynamics yf(t) should thus provide robust estimates of kpm/h (for a known Fd) at long time scales, since it does not depend on the value of J0. Finally, no clear quantitative comparisons have been done to our knowledge between eqn (3)–(6) and experimental data. Importantly, t0 and y0 are related to each other as y0/t0 ¼ J0F0/DF, and any comparison with the experiments should not take these two parameters as independent tting parameters. Modied model taking into account Kelvin's effect
During the drying process, the pressure drop dp due to the viscous dissipation in the solid region, continuously increases until the receding of the drying interface. At this stage, dp ¼ pm reaches very large values for small colloids as those investigated in ref. 9, 13 and 14 and in the present work (a z 6–13 nm): pm z This journal is © The Royal Society of Chemistry 2014
where pk is the Kelvin pressure given by: pk ¼ kBT/vm z 1380 bar,
(8)
at room temperature (vm is the molecular volume of water). For radii a ¼ 6–13 nm, one has indeed J/J0 z 0.3–0.6 (J/J0 z 0.6–0.8 for pm ¼ 5.3g/a). We thus modied the model described by Wallenstein and Russel20 to take into account this effect. Before air invasion (and thus for dp < pm), the compaction front yf and the evaporation rate J follow the relationships: DFy_ f ¼ F0J, J ¼ J0 exp(dp/pk), dp ¼ hJyf/k.
(5)
We add a more complete derivation of this relationship in the ESI.† The dynamics of the compaction front can then be estimated according to the conservation equation (eqn (2)) leading to the square root behavior:20 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yf ¼ y0 1 þ 2ðt t0 Þ=t0 : (6)
2.2
720–1550 bar for pm ¼ 2g/(0.15a) (pm z 280–620 bar for pm ¼ 5.3g/a). For such huge pressure drops, one cannot neglect the decrease of the chemical potential of water at the drying interface due to the Kelvin law.23 It indeed yields a decrease of the evaporation rate according to: dp J ¼ J0 exp ; (7) pk
The possibly high viscous dissipation in the solid region thus decreases J(t), and consequently slows down the growth rate y_ f. We introduce the natural length scale yk ¼ kpk/(hJ0), and get the unitless equations: d~ yf ~ ¼ J; dt~ J~ ¼ exp(J~y~f),
(9)
(10)
with ~yf ¼ yf/yk, ~t ¼ t/tk, and ~J ¼ J/J0, where tk ¼ (DF/F0)(yk/J0). For small pressure drops compared to pk, one can linearize the exponential term, and nd simply the dynamics of the compaction front: sffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2t yf z yk 1þ 1 ; (11) tk Jz
J0 J0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: 1 þ yf yk 1 þ 2t=tk
(12)
Interestingly, this equation has a similar form ffias the one pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi found empirically in ref. 9 (yf ¼ y0 ð 1 þ 2t=t0 1Þ; see previous section), but with the important difference that yk a2, as compared with y0 that scales linearly with a. This difference may explain the observations of ref. 9, i.e. measurements of y0 for several a, which deviate from a linear scaling. The regime described above fails as soon as the pressure drop dp reaches its maximal value pm corresponding to the receding of the air–water interface. Again, there is an additional mass transfer resistance, and following the arguments provided by Wallenstein and Russel:20 Jyf z Jiyi ¼ kpm/h, where Ji < J0 is
Soft Matter, 2014, 10, 4151–4161 | 4153
View Article Online
Soft Matter
Published on 18 March 2014. Downloaded by University of Illinois at Chicago on 26/10/2014 07:17:30.
the evaporation rate when dp ¼ pm, and yi the corresponding position of the compaction front. At longer time scales, the dynamics follows again: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F0 Ji yf ¼ yi 1 þ 2 ðt ti Þ; (13) DFyi with ti given by yf(ti) ¼ yi. In the following, we use numerical solutions of eqn 9 and 10 to compute yf(t) and J(t) before the receding of the interface. Fig. 2 displays the dynamics of the compaction front and of the evaporation rate for a ¼ 13 nm and a ¼ 6 nm (pm ¼ 2g/ (0.15a)), for both models, i.e. with and without Kelvin's effect. At long time scales (t [ tk), both models lead to close dynamics for yf(t) and J(t). In this regime indeed, evaporation is mainly governed by the highly resistive mass transfer of vapor through the pores of the dry solid. At an early stage however, a clear difference is observed, mainly for the temporal behavior of J(t) (the difference between the two dynamics of yf(t) is less pronounced). Particularly, no constant J(t) is observed when taking into account Kelvin's law, and the receding of the drying interface occurs later due the initial decrease of J(t) because of the increase of the viscous dissipation. These behaviors depend strongly on the size of the colloids (as well as the numerical estimates of pm), and the differences between the two dynamics are more important in the case of small colloids, see the insets of Fig. 2 for a ¼ 6 nm. For a >
Paper
15 nm, the two different compaction front curves are almost indistinguishable, but a signicant decrease of J can always be observed at small time scales when taking into account the Kelvin effect (data not shown). In the following, we try to compare these predictions (along with the model of ref. 20) with experimental data, and experimental values inferred from recent studies.9,10,13,14
3 Comparison with experimental data 3.1
Measurements from the literature
Surprisingly, no clear attempts have been made to our knowledge to compare quantitatively experimental data to the theoretical predictions given above. The square root dynamics pffiffi yf t has been clearly identied in ref. 9, 10, 13 and 14, howeverpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi only ffi ts based on the empirical law yf ¼ y0 ð 1 þ 2t=t0 1Þ are provided, and more importantly with no clear demonstration that both y0 and t0 are not two independent parameters. Such a quantitative comparison should a priori lead to numerical values of the permeability k of the solid. Indeed, at long time scales, i.e. for a receding drying interface, measurements of both J(t) and yf(t) should give precise estimates of kpm as Jyf z kpm/h; the same measurements at an early stage may discriminate the two models given above (Fig. 2). Models and experiments that try to understand the formation of fractures and delamination during unidirectional drying use the Carman–Kozeny relationship: k ¼ a2
ð1 Fd Þ3 ; 45Fd 2
(14)
to compute the permeability k of the solid region. The latter is known to describe very well the ow through randomly closepacked monodisperse spheres,25 but despite an extensive literature survey, we do not nd any conrmation of the validity of this law for the small colloids typically investigated in the present work and in ref. 9, 10, 13 and 14 (silica particles, a < 15 nm). Such direct measurements are indeed probably difficult to perform due to the very high pressure drops involved, and the possible formation of fractures that can lead to experimental artifacts. Table 1 Experimental parameters corresponding to unidirectional drying experiments reported in ref. 10, 13 and 14 indicate values that are not measured (or not reported in the corresponding reference)
Fig. 2 Top: (a ¼ 13 nm) compaction front yf normalized by yk; model of ref. 20, and model taking into account Kelvin's effect. Both symbols indicate the transition to the regime of the receding of the drying interface. The dotted line indicates the linear growth given by eqn (3). Bottom: corresponding normalized evaporation rates J/J0. The two insets show the same plots but for a ¼ 6 nm.
4154 | Soft Matter, 2014, 10, 4151–4161
Ref.
Fig. 2 of ref. 10
Fig. 3 of ref. 13
Fig. 4 of ref. 14
Radius a F0 Fd J0 (mm s1) Cell (mm mm) y0 (exp.) y0, eqn (4) t0 (exp.) t0, eqn (4)
11 nm 0.2 0.58 1a 0.1 2 1 mm 5 (2) cm
