Capillary rise dynamics of aqueous glycerol solutions in glass capillaries: A critical examination of the Washburn equation

Journal of Colloid and Interface Science 411 (2013) 257–264

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Capillary rise dynamics of aqueous glycerol solutions in glass capillaries: A critical examination of the Washburn equation M. O’Loughlin, K. Wilk 1, C. Priest ⇑, J. Ralston, M.N. Popescu * Ian Wark Research Institute, University of South Australia, Adelaide, SA 5095, Australia

a r t i c l e

i n f o

Article history: Received 6 February 2013 Accepted 31 May 2013 Available online 7 June 2013 Keywords: Capillary rise Washburn equation Dynamic contact angle Glycerol–water mixtures

a b s t r a c t The classic description of capillary rise given by the Washburn equation was recently questioned in the light of experimental evidence for a velocity dependent dynamic contact angle at a moving contact line. We present a systematic investigation of the capillary rise dynamics of glycerol and aqueous glycerol solutions in vertical glass capillaries of various radii. For pure glycerol, the results of our experiments are in almost perfect agreement with the predictions of the Washburn equation using independently measured values for the liquid and capillary parameters. For aqueous glycerol solutions we observe discrepancies between the theoretical expectations and the experimental results, which are increasing with the water content of the solution. A thorough analysis, combined with scaling arguments, allows us to conclude that dynamic contact angle effects alone cannot provide a consistent explanation for these discrepancies. Rather, they can be perfectly accounted for if the mixture flowing in the capillary would have an effective, increased viscosity (in respect to the nominal value). We suggest and briefly discuss various mechanisms that could contribute to this observed behavior. Ó 2013 Published by Elsevier Inc.

1. Introduction The capillary penetration of a wetting liquid into a narrow tube or slit-like channel in contact with bulk liquid is a ubiquitous natural phenomenon. The rise of underground water in soil, water transport from the roots to the leaves of a plant, and the spreading of a coffee stain in table cloth are some of the everyday life examples of capillary-driven wetting [1]. Moreover, a wide variety of technological applications such as oil extraction through porous rocks, copper extraction via acid leaching, or ink-jet printing, are fundamentally dependent on capillary flows. With the advent of modern micro- and nano-scale technologies capable of engineering channels of very small cross-sectional dimensions, the question of capillary driven flows in micro- and nano-channels has been recently approached in a number of papers [2–10]. Interest in this subject, from engineering, applied science, and theoretical perspectives, has repeatedly resurfaced during the time since the first theoretical description of such flows, the Washburn equation, was proposed [11–13]. It is impossible to overstate the importance of the Washburn equation; it has found applications in a variety of research questions, such as determining the wetting properties (contact angle) of fine particles [14–18], development of

⇑ Corresponding authors. E-mail addresses: [email protected] (C. Priest), Mihail.Popescu@unisa. edu.au (M.N. Popescu). 1 Present address: Oil and Gas Institute,Armii Krajowej 3, 38-400 Krosno, Poland. 0021-9797/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jcis.2013.05.077

viscometers [19,20], or the characterization of the wetting properties of general porous media [21–27,2–4]. A large number of studies have addressed various aspects of capillary driven flows in channels with complex geometries, such as the influence of the shape of the channel’s cross section on the shape of the meniscus and on the dynamics of the capillary imbibition [28–37] or the flows in channels with variable cross-sections [38–41]. The question of capillary imbibition from a finite (drop), rather than infinite (liquid bath), reservoir into a porous material [42–45,27,46] or into a single cylindrical capillary [47–53], as well as that of the motion of small liquid plugs (‘‘slugs’’) inside capillaries [54,55] has been also extensively discussed in the literature. Furthermore, recently there has been increased theoretical interest in the capillary driven flows involving non-Newtonian liquids [56] or capillaries with elastic [57] or heterogeneous, either topologically [58–60] or chemically [61–63], walls. Because even for the simplest geometries of the channel, like a cylinder or a slit between parallel plates, the derivation of Washburn’s equation relies on a number of assumptions that sometime are difficult to directly asses (see also Appendix A), the extent of the validity of Washburn equation has been the subject of numerous experimental investigations [64,65,1,66– 69], numerical simulations [70,71,62,72–80], and theoretical studies aiming at elucidating the influence on the dynamics of end effects [1,81], of flow effects due to the meniscus shape [82], as well as of liquid inertia terms in the dynamic equations [65,83–86].

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The issue of the influence of a velocity v dependent contact angle h [87,54,88–91] on the rate of the capillary rise has been approached in a number of studies [92,93,55,72,94–106]. By analogy with the classical derivation of Washburn [11], the resulting dynamics h(t), where h is the height of the liquid column in the tube at time t, derived under the assumption of a capillary pressure determined by the instantaneous, dynamic contact angle has been called a ‘‘modified’’ Washburn equation. Several experimental studies [92,93,98] and numerical simulations investigations [72,94] reported deviations from the predictions of the classic Washburn equation and good agreement with modified ones based upon rather different expressions for the velocity dependence of the dynamic contact angle. However, none of them thoroughly discussed why such deviations from the classical result of Washburn have not been previously observed. For example, Ref. [68] concluded that Washburn equation provides an accurate description of the experimental results for capillary driven flows in horizontal capillaries. Similar conclusions of validity of the classic Washburn equation are drawn from MD simulations in Refs. [62,107]. In a recent publication [105], we have discussed this apparent dilemma. We have shown that for a number of liquids the theoretically predicted deviations from the classic Washburn equation due to the most common possible choices h(v) for the velocity dependence of the contact angle [88–90,92] are very small, likely within the experimental error bars, and thus undetectable. The contradictory experimental and numerical findings discussed above indicate that the ‘‘simple’’ question of the limits of validity of the Washburn is far from being settled and that systematic, thorough studies are yet needed in order to provide a clear and comprehensive answer to it. We report here experimental results for the capillary rise dynamics of glycerol and aqueous glycerol solutions in glass capillaries of various radii vertically inserted in the liquid. A thorough analysis of the results, combined with scaling arguments, leads to the conclusion that, when observed, deviations from the predictions of the Washburn’s equation cannot be consistently explained solely by dynamic contact angle effects. We show that the observed discrepancies could rather be consistently accounted for, without the need to modify Washburn’s equation, by an effective, increased viscosity (in respect to the nominal value) of the mixture flowing in the capillary. Various mechanisms, assuming a preferential adsorption of water at the capillary walls, that could contribute to this observed behavior are suggested and discussed. The organization of the paper is as follows. Section 2 is devoted to a thorough discussion of the experimental set-up. In Section 3 we discuss and interpret, based on the classic and ‘‘modified’’ Washburn equations, respectively, the results of the capillary rise studies (a brief overview of the mathematical details is provided in Appendix A). The conclusions of this study are presented in Section 4.

2. Materials and methods 2.1. Capillaries Precision borosilicate glass capillaries were purchased from Friedrich and Dimmock. For the studies presented in this paper we have employed capillaries with various inner radii r between 0.465 mm and 0.215 mm. The inner radii were measured for each capillary, at both ends, by using optical microscopy; this, rather than using the manufacturer’s quoted value, ensured the accuracy of the further data analysis. For each batch, we have randomly chosen a few capillaries and additionally measured the values of the radius along the length of the capillary via fracture of the capillary. We found no significant differences between measurements at

various points (up to ±3 lm, which is less than 1–2% for the capillaries employed); therefore we infer that the capillaries are indeed of constant inner radius along the whole length. Using the fractured pieces, we have also verified the smoothness of the inner surface of the capillaries (see also Ref. [108]). The imaging of the surface has been done using a NanoScope III atomic force microscope (AFM; Digital Instruments, Santa Barbara, CA), with ultrasharp silicon SPM cantilevers (NT-MDT, Moscow), in tapping mode. We have found that the surfaces are very smooth, with a root-mean-squared roughness (calculated with the AFM software) of only 0.35 nm over an area of 1 lm2 [108]. 2.2. Water and general chemicals Ultra-pure water (pH of 5.6 ± 0.1, resistivity of 18.2 MX cm, and surface tension 72.8 mN/m at 22 °C) was obtained using a MilliQ Element-Millipore system. Glycerol (analytical reagent (A.R.), 99.5%) and potassium hydroxide (90%) were purchased from Chem-Supply Australia. Glycerol solutions of desired concentrations (by weight) were freshly prepared before experiments by using an electronic balance (OHAUS E4000). Known masses of ultra-pure water and glycerol were added in a clean glass container. For example, a 70% (by weight) glycerol solution was prepared by combining 28 g of glycerol with 12 g of ultra-pure water (40 g total). To improve the optical contrast, a small amount of red dye (Allura Red AC) was added to the solution. The solutions were mixed by hand shaking for 1 min followed by 15 min of stirring using a magnetic stirrer (IEC). 2.3. Cleaning The glassware was cleaned in a caustic bath using a warm solution of 2 M potassium hydroxide for 1 h. It was then washed thoroughly with ultra-pure water and placed in an oven for 2 h at 110 °C to dry. One end of the capillary was chosen for immersion in the liquid and was therefore not touched at any stage of cleaning or during the experiment. To clean the capillaries before experiments, 8 ml of 2 M potassium hydroxide was flowed through the capillary at a rate of 1 ml/min using a syringe pump (K D Scientific, KDS 210P). This was followed by two steps of rinsing, each with the same amount (8 ml) of ultra pure water at the same 1 ml/min volumetric rate. After removing the capillary from the tubing, an additional brief rinsing of the whole capillary (inside and outside) under the high pressure ultra-pure water flow of the MilliQ system was performed before the capillaries were dried with filtered N2 gas (99.9%, BOC Australia). The capillaries were then immediately used in a capillary rise experiment. The MilliQ system was also used to rinse the glycerol from the capillary between sub-sequent experiments. This was followed by two steps of rinsing with ultra pure water using the syringe pump (8 ml at 1 ml/min volumetric rate in each step). For the smallest capillary (r = 0.215 mm) this latter step was repeated if the surface was not sufficiently cleaned, i.e., if the equilibrium capillary rise indicates an equilibrium contact angle of more than 10°. 2.4. Physical properties of the aqueous glycerol solutions The surface tension (liquid–vapor) and the density of glycerol and aqueous glycerol solutions were measured using a Wilhelmy Balance (DCAT 21, Data Physics) at room temperature (22 ± 1 °C) using a platinum iridium plate (10 mm  19.9 mm  0.2 mm) and a silicone probe (volume 1.351 cm3, mass 3.146 g, and thus density of 2329 kg/m3). The viscosities of glycerol and aqueous glycerol solutions at room temperature (22 ± 1 °C) were measured with a Brookfield viscometer (DV-11 Pro) using a spindle (LV-2)

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rotating at 20 rpm. The results of these measurements are listed in Table 1. We note that they fall within the range of values reported in the literature [109–113]. We also note that these properties have been measured for the liquids with or without the red dye added and no differences have been observed between the two cases. 2.5. Capillary rise measurements All capillary rise measurements were carried out in a Class 1000 clean room at 22 ± 1 °C. Taking advantage of the fact that glass capillaries are transparent, we followed the capillary rise using optical microscopy. Fresh glycerol or aqueous glycerol solution was poured in a clean Petri dish, slightly over filling it (see Fig. 1); this allows imaging immediately after the capillary touches the liquid, while avoiding optical distortion (which would occur if we chose to image through the Petri dish). The liquid was brought into contact with the capillary by control of a motorized stage (motor speed 1 lm/s). The capillary rise dynamics was recorded using a standard speed camera Dino-Lite, Digital Microscope Pro (capture rate of up to 15 frames per second), which allowed us to follow the capillary rise from the early stages. Individual frames from the recorded movie were then analyzed to determine the capillary rise height h(t) using ImageJ software [114]. The pixel size calibration was based on the known outer diameter of the capillary (independently measured prior to experiments), and the time (measured from the first frame in which a change in the shape of the surface of the liquid bath at the contact with capillary was detected) was obtained from the known frame rate of the recording. 3. Results and discussion The classic description of the capillary rise dynamics in a vertical capillary was given by Washburn [11] as the balance between the driving force provided by the capillary pressure and the retarding effects of the weight of the liquid column and of the viscous resistance of the flowing liquid column. The resulting Washburn equation for the time t for the liquid column to rise to the height h is given by [11]

 tðhÞ ¼ T

  h h ; þ cos he ln 1  L L cos he

ð1Þ

r and T ¼ 16rg (see Eq. (A.3)) are characteristic length where L ¼ q2gr q2 g 2 r3 and time scales that depend on the liquid (viscosity, density, and surface tension) and on the radius of capillary. At short times, 2

h ðtÞ ’

L2 2 cos he T

! t

  rr cos he t; 4g

ð2Þ

while asymptotically at long times approaches the equilibrium rise he

hðt ! 1Þ ! he ¼

2r cos he : qgr

ð3Þ

Through the usual procedure of equating the expression for h(tc) in Eq. (2) with he, one defines a characteristic time tc for the crossover between the two regimes,

Table 1 Surface tension r, density q, and viscosity g of glycerol and aqueous glycerol solutions at 22 ± 1 °C. Glycerol %

r (mN/m)

q (kg/m3)

g (Pa s)

99.5 85.0 70.0

63.8 ± 0.1 64.5 ± 0.1 65.3 ± 0.1

1255 ± 1 1221 ± 1 1181 ± 1

1.11 ± 0.01 0.11 ± 0.01 0.02 ± 0.01

Fig. 1. Experimental set-up for capillary rise measurements, showing the Petri dish over-filled with glycerol (containing red dye for good optical contrast) and the capillary in contact with the liquid bath. The capillary rise h at time t is measured from the surface of the liquid bath to the bottom of the meniscus. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

tc ¼

8r cos he 3 gr : q2 g 2

ð4Þ

For the liquids employed in our study, the variations in r and q, as well as in cos he (expected to be close to unity in all cases) are very small compared to the variations in the viscosity. By using capillaries of various radii we achieve significant variations in the range he of the capillary rise, while the crossover time tc is significantly varied both through the variations in the viscosity (various liquids), at fixed radius, as well as through variations in the capillary radii at fixed viscosity (one liquid). Fig. 2a–c presents the experimental results for h(t) (symbols) together with the corresponding (i.e., the corresponding L and T values) theoretical predictions from Eq. (1). The value used for the contact angle, which is the only fitting parameter, is he = 0 in all cases. It can be seen from Fig. 2a–c that this choice leads to almost perfect agreement between the theoretically predicted, Eq. (3), and the experimentally measured equilibrium rise he = h(t ? 1). (We note, however, that any value he [ 10° – as expected for glycerol and aqueous glycerol solutions on clean glass [110]) – would produce similar results. Excellent agreement between experiment and theory, Eq. (1) without any adjustable parameter (once the value of the contact angle was fixed to he = 0) is observed for glycerol, for all capillaries, and very good agreement is observed for the 85% glycerol solution, although on close inspection some small deviations are observed in the region of the transition towards the asymptotic he value. For the 70% glycerol solution, the deviations from the theoretical predictions in the transition region, especially for the smaller radii capillaries, are significant. The very good reproducibility of the h(t) curves upon repeating the experiments, as well as the value he [ 10° and the excellent data collapse in the asymptotic (s  1) regime, are strong indications of clean and smooth inner surfaces of the capillaries, as well as of no noticeable contamination of the liquids or capillaries during the experiment. We therefore conclude that the observed deviations, which furthermore show a somewhat systematic trend (more significant at lower concentration of glycerol), cannot be spurious. To understand these deviations, and to be able to assess their origin in a consistent manner, we proceed with a more detailed analysis of the experimental data.

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(a)

(a)

(b)

(b)

(c)

(c)

Fig. 2. Experimental data (symbols) and corresponding theoretical predictions (lines) from Eq. (1) for (a) pure glycerol; (b) 85% and (c) 70% glycerol solutions, respectively. For each capillary radius, three independent measurements are presented (squares, circles, and triangles, respectively). The insets are zoom-ins of the dynamics at early stages.

Assuming that Eq. (1) provides a valid description for the dynamics of capillary rise, it follows that when data is re-plotted in dimensionless form as v = h/L as a function of s = t/T all the experimental curves (for all liquids, and for all capillaries) should collapse onto the master curve, Eq. (A.5). As can be seen in Fig. 3a–c, this indeed is the case for each liquid: very good collapse of the data for capillaries of various radii is observed. As shown in the insets, the quality of the data collapse at the early stages of the capillary rise is affected by a certain amount of scattering. This scatter is most likely an indication of the limited accuracy of the

Fig. 3. Dimensionless v = h/L as a function of s = t/T experimental data for (a) pure glycerol; (b) 85% and (c) 70% glycerol solutions, respectively. For each capillary radius, three independent measurements are presented with the same symbol. The insets are zoom-ins of the dynamics at early stages.

experiment (for example, small variations in the actual value of the contact angle, between 0° and 10°), as can be inferred from the spread in data from measurement to measurement in the case of glycerol in the capillary of radius r = 0.215 mm (the three sets of blue triangles in Fig. 3a).2 However, when plotted also for all liquids together, the systematic deviations from scaling with decreasing glycerol concentration, which have been noted in the discussion of

2 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.

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where f denotes a so-called ‘‘coefficient of wetting-line friction’’ (note that it has units of a surface viscosity). In terms of the parameters k0 (‘‘frequency of molecular jumps’’) and k (‘‘average distance between adsorption sites’’) of the MK model, f is expressed as

f¼

kB H k0 k3

ð7Þ

;

where kB is Boltzmann constant and H denotes the absolute temperature. The corresponding modified Washburn equation is obtained by replacing in Eq. (A.1) the equilibrium contact angle by the dynamic one, cos he ? cos h, and then substituting cos h with the expression in Eq. (6). After separating the variables and integrating, one obtains the modified Washburn equation (in nondimensional variables) [105]



Fig. 4. Dimensionless experimental v = h/L as a function of s = t/T data (symbols) for all liquids and all capillary radii. The black solid line corresponds to the theoretical prediction, Eq. (A.5). The red and blue lines correspond to the ‘‘viscosity corrected‘‘ Eq. (11), where the correction factors are p = 1.2 and 1.6 (thereby an effective increase in viscosity) for the 85% and the 70% glycerol solutions, respectively. The inset is a zoom-in of the dynamics at early stages. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2, are now clearly evidenced (see Fig. 4). As we discuss below, these differences in the scaling behavior provide the means to discriminate between a priori plausible scenarios. One possible cause of deviations from the behavior predicted by the Washburn equation is that of dynamic contact angle effects. Within the description of dynamic contact angle ‘‘modified’’ Washburn equation, the capillary rise will always proceed slower than the one predicted by the Washburn equation, because the dynamic contact angle for an advancing contact line is presumably always larger than the equilibrium one. At first glance, the deviations we observe are qualitatively compatible (a slower dynamics) with dynamic contact angle effects. However, as we show below, none of the commonly employed models for the velocity dependent contact angle [89,88] leads to predictions compatible with both the scaling behavior for a given liquid (Fig. 3) and the deviations from scaling for various liquids (Fig. 4). We first discuss the so-called ‘‘hydrodynamic’’ model [88], for which the velocity-dependence of the dynamic contact angle is given by

GðhÞ ¼ Gðh0 Þ þ c

g dh gV dv  Gðh0 Þ þ c ; r dt r ds

ð5Þ

where G is a complicated (but system independent) function and c is a constant (or very weakly dependent on the system, as the logarithm of a slip length); for the following argument, the exact expression of the function G is not needed. Since gV is independent of g (see Eq. (A.4)), in scaled variables the magnitude of the velocity correction (last term in Eq. (5)) becomes independent of g. Therefore this model cannot explain deviations which are viscosity dependent, as observed in the experiment. We now consider the second usually employed model, the Molecular Kinetic (MK) model of Blake [89]. For clarity, we discuss first the case of the linear version of the model. This choice leads to a particularly simple form of modified Washburn equation, which thus reduces the cumbersome algebra and allows for very clear, physically intuitive arguments. The dynamic contact angle h as a function of the meniscus velocity dh/dt is given by

cos h ¼ cos he 

f dh

r dt

;

ð6Þ

sB ðvÞ ¼ sðvÞ  A ln 1 

v cos he

 ;

ð8Þ

where sB(v) denotes the (dimensionless) time at which the (dimensionless) capillary rise is v, s(v) is given by Eq. (A.5)

A¼f

qgr2 ; 8r g

ð9Þ

i.e., the deviations from the classic Washburn equation are additive. The crucial point is that the magnitude of these deviations, which is characterized by A, depends both on the liquid properties and on the geometrical characteristics (the radius of the capillary). Therefore the scaled representation v(s) would not lead to data collapse, but at most to an apparent one if the magnitude of the deviations is sufficiently small. The data collapse for capillaries with different radii, as shown in Fig. 3a–c, implies that this apparent scaling is the case here. Thus the corresponding values of A should be such that for each liquid they satisfy maxr A [ 0.1 (see Ref. [105] and Fig. 5 therein), where maxr denotes the maximum in respect to r, over the range of capillary radii r explored in the experiments. However, the argument above now also implies that for any of the liquids and for any of the capillary radii the magnitude A [ 0.1. Consequently, there should be no observable deviations from data collapse also in the case when v(s) is plotted for the various liquids. This is in conflict with the experimental findings shown in Fig. 4; therefore, dynamic contact angle effects alone cannot provide a consistent explanation for the experimental results. Rather, the inequality maxr A [ 0.1 applied for each liquid (noting that the maximum occurs for the largest capillary employed in the corresponding set of experiments) gives bounds on f (and then on k0, since 1 A  f  k0 , by assuming that in all cases k ’ 1 nm), see Eqs. (9) and (7). Such lower bounds are listed in Table 2; we note that these rough estimates of the k0 bounds do match (order of magnitude) the typical values reported in the literature (see, e.g., Refs. [115,91]). For the general MK model, the velocity dependency of the dynamic contact angle is given by the implicit formula [89]

" # dv 2k0 k brk2 ¼ sinh ðcos he  cos hÞ ; V ds 2

ð10Þ

Table 2 Upper bounds for the coefficient of wetting-line friction f and corresponding lower bounds for the frequency k0 of molecular jumps implied by maxrA [ 0.1. Glycerol %

f(max) (Pa s)

k0

99.5 85.0 70.0

21.3 2.45 0.47

0.19 1.66 8.75

ðmaxÞ

(MHz)

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where b = 1/(kBH). Noting that for our systems one does not expect the value of k to vary significantly from liquid to liquid, the prefactor brk2 inside the sinh() function is basically a system independent constant. Therefore the velocity dependence and, consequently, the magnitude of the deviations from the predictions of the classic Washburn equation are determined solely by the factor

B¼

2k0 k 16k k0 g ¼ : V qg r2

From here the line of reasoning, based on the argument that the lack of deviations from scaling in Fig. 3a–c requires for each liquid that minrB must be larger than some threshold, is similar to the one developed for the linear approximation of the MK model. (It can be checked numerically that for B J 1 apparent scaling holds, and the lower bounds on k0 inferred from this inequality are close to the ones obtained in the linear MK approximation.) Therefore we reach the same conclusions, that dynamic contact angle effects alone cannot provide a consistent explanation for the experimental findings shown in Figs. 3 and 4. Finally we note that: (i) based on similar arguments, it can be easily shown that the heuristic expression for the velocity dependent dynamic contact angle proposed by Joos et al. [92,93] is incompatible with the experimental findings; and (ii) it was not possible to carry-out the above analysis when employing the velocity dependent dynamic contact angle model proposed by Shikhmurzaev [90,115]. This is because the model depends in a complicated manner on a number of (possibly system-dependent) parameters and little is known about their exact dependence on the viscosity of the liquid and possibly on the radius of the capillary. However, in view of the fact for a number of systems this model leads to predictions similar to those of the MK model [115,105], we expect that it will not lead to a different picture in the present context either. The arguments so far rule out contamination or dynamic contact angle effects as the source of discrepancies between experiment and the Washburn equation. The thorough experimental preparation also rules out inaccuracy of the concentration of prepared solutions, temperature variations (checked during experiments), or significant water evaporation from the solutions (checked through independent additional measurements) as possible sources of the discrepancies observed for the glycerol solutions. Since the data collapse for each liquid (considered separately) is a very strong argument for Washburn-like scaling, we explore below the hypothesis that perhaps the observed slower dynamics is compatible with that due to an apparent, effectively increased (from the independently measured bulk value) viscosity of the flowing mixture during the rise in the capillaries. Consider that the effective viscosity of the flowing solution is actually g0 = pg, where p is the correction factor and g the independently measured bulk value of the viscosity. It is easy to see that the calculations in Appendix A are changed by replacing: T ? pT, and thus s ? s/p (L and v remain unchanged), which leads to the modified theoretical prediction





sp ðvÞ ¼ p v  cos he ln 1 

v cos he



 p sðvÞ:

ð11Þ

As shown in Fig. 4 (the solid red and blue lines), this scenario can indeed restore the perfect agreement between the experiment and theory. However, this is achieved at the expense of accepting significantly (at first glance) increased effective viscosity values, by factors of 1.2 and 1.6 for the 85% and 70% mixtures, respectively. The question then arising is what may cause such an apparent, increased viscosity. It is known that the viscosity of glycerol aqueous solutions with large glycerol content strongly depends on the temperature and

composition [109–113]. We first note that variations in temperature alone cannot consistently explain all the deviations observed. An effective increase in viscosity by factors of 1.2 and 1.6 for the 85% and 70% mixtures would require temperatures lower by 2 and 10 °C, respectively. The latter is simply impossible to have happened under the conditions of the experiment. Therefore below we focus on the dependence on composition. We suggest that one possibility is that there is strong preferential adsorption of water at the glass walls, which would effectively lead to a water-enriched region near the wall in contact with a water-depleted solution flowing through the capillary. This hypothesis is motivated by the well known strong affinity of water for silica surfaces, evidenced through the formation of strongly bound thick wetting layers [116–119]. However, even for the strong dependence of the viscosity of glycerol solutions on the composition [109–113], the factors p = 1.2 and 1.6 observed in our experiments would correspond to effective glycerol content of ’87%, instead of the nominal 85%, and ’75%, instead of the nominal 70%, respectively. Noting that for our capillaries the latter case would imply a few microns value for the thickness of the depleted layer (even under the assumption of complete demixing), it seems likely that considering just the motion of a water-depleted core may account only partially for the deviations observed. Only by including in the description the necessary re-circulating radial flows between the water-depleted core and water-enriched layer at the wall it may become possible to account for the full magnitude of the deviations. We note that the formation of a water-enriched (glycerol-depleted) region near the walls of the capillary bears similarities with the phenomenology observed in the capillary rise of surfactant solutions [120]. In this latter case, a depletion of surfactant occurs at the advancing meniscus and leads to a slowing down of the capillary rise due to coupling of the flow with the mass-transport of the surfactant from the bulk towards the meniscus. This offers an alternative mechanism for an effective, apparent increased viscosity that can restore the agreement between the experimental observation and the theoretical predictions of the classic Washburn equation (which does not account for inhomogeneities in the composition of the flowing liquid). In this context it is important to note that water adsorption at the capillary wall may contribute to the observed slower than expected dynamics of the mixture also through the formation of precursor films of water ahead of the advancing meniscus (in similarity with a mechanism revealed in the thin-film wicking studies, involving various liquids, of Chibowski and Hołysz [121–124]). For water on glass, the dynamics of formation of such films is rather fast, and thus it could lead to a water depletion of the mixture during the early stages of the capillary rise. These discussed mechanisms remain, however, at the stage of speculative suggestions. Further investigations are required to elucidate which of them (or maybe a completely different scenario) is the main cause for the discrepancies observed and if the effects of such mechanisms could be accounted for via an effective viscosity in the classic Washburn description, as it is seemingly the case of our systems, or more complex modifications, as in the case of surfactant solutions [120,1], are necessary. 4. Conclusions In conclusion, we have performed a systematic investigation of the capillary rise dynamics of glycerol and aqueous glycerol solutions in vertical glass capillaries of various radii. While for pure glycerol the experimental results are in very good agreement with the theoretical predictions of the Washburn equation, for aqueous glycerol solutions we have noted deviations, which are increasing with the water content of the solution, from the theoretical

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expectations. In contrast with previous reports, where such deviations have been proposed as evidence for break-down of the classic Washburn equation description (see, e.g., Refs. [92,93,98,103]), we conclude here, based on a thorough analysis of the scaling behavior of the capillary rise h(t) for various liquids and capillaries, that the deviations we observed cannot be consistently explained by invoking dynamic contact angle effects. We suggest that they are rather due to an effective, increased viscosity of the glycerol solutions. We argue that this may occur, e.g., via water-depletion due to preferential adsorption of water at the glass walls or formation of precursor films ahead of the advancing liquid column, which may induce recirculating mass transport currents effectively slowing down the capillary rise, in similarity with the behavior observed for surfactant solutions. Further investigations employing a similar thorough data analysis and various other simple liquids, as well as mixtures, and materials for capillaries (such that r, he, as well as the molecular structure of the liquid, are varied in a wide range) are obviously needed before a complete picture can be drawn. Acknowledgments Financial support from the Australian Research Council (ARC) and AMIRA International via the ARC Linkage Grant LP0667828 (AMSRI) and the ARC Discovery Grant DP1094337 is gratefully acknowledged. The authors thank the anonymous reviewers for their insight into the behavior of glycerol-water mixtures and the similarities with the dynamics of surfactant solutions. Appendix A. Classic Washburn equation

(i) the extent tt of the transient hydrodynamic regime after the start of the capillary rise is very short compared with the time-scale of the capillary rise [65,1]. For all t > tt; (ii) the contact angle h of the advancing meniscus has the constant value he; (iii) the relaxation of the velocity profile toward a Poiseuille– Hagen flow corresponding to the pressure drop at time t between the advancing meniscus and the bottom of the capillary can be considered as almost instantaneous; (iv) the vapor phase has negligible hydraulic resistance; (v) any end effects associated with the finite length of the tube and the presence of a meniscus are negligible [11,65]. Within the confines of these assumptions, the balance between the driving force provided by the capillary pressure, the weight of the liquid column, and the viscous resistance leads to the following dimensionless differential equation for the rate of capillary rise:

dv ¼ cos he  v; ds

Glycerol %

L (mm)

T (s)

V (mm/s)

99.5 85.0 70.0

48.2 50.1 52.4

752.2 79.6 15.7

0.06 0.63 3.35

L¼

2r 16rg ; T ¼ 2 2 3; qgr qgr

ðA:3Þ

capillary rise and the time of rise, respectively. Note that L would be the equilibrium rise for a complete wetting case (he = 0), and thus v 6 cos he 6 1. Additionally, L/T defines the characteristic velocity

V ¼ L=T ¼

qgr2 : 8g

ðA:4Þ

For the narrowest capillary, of radius r = 0.215 mm, used in this study, the values of these characteristic length, time, and velocity scales corresponding to the various glycerol solutions are listed in Table A.0. For any of the other capillaries, of radius r0 , the L,T, and V values can be obtained from the corresponding ones in Table A.0 by multiplying them with factors (r/r0 ), (r/r0 )3, and (r0 /r)2, respectively. Using the initial condition v(0) = 0, Eq. (A.1) is solved for s:



sðvÞ ¼ v  cos he ln 1 

v cos he

 ;

ðA:5Þ

which is the classical Washburn equation.3 By replacing s = t/T and

The derivation of Washburn equation [11] for the capillary rise dynamics h(t) (where t = 0 is the time when the liquid enters in the capillary and h = 0 is the level of the liquid in the bath, see Fig. 1) in a uniform cross section, cylindrical, open capillary of radius r, with chemically homogeneous, inert walls, inserted vertically in a bath filled with an incompressible, non-volatile, Newtonian liquid of density q, viscosity g, liquid–vapor surface tension r, and equilibrium (obeying Young’s equation) liquid–solid contact angle he is based on the following assumptions (for a detailed discussion see, e.g., Refs. [11,65,1])

v

Table A.0 Length, time, and velocity scales corresponding to the liquids used in this study in a capillary of radius r = 0.215 mm.

ðA:1Þ

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where

v ¼ h=L; s ¼ t=T are the dimensionless, i.e., measured in units of

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3 It is easy to see that s(v) is continuous and monotonically increasing for v P 0; therefore, it can be inverted for v(s), s P 0 (see also Ref. [101]), and v(s) is also a continuous and monotonically increasing function.

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