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Spreading and Arrest of a Molten Liquid on Cold Substrates Faryar Tavakoli, Stephen Davis, and H. Pirouz Kavehpour Langmuir, Just Accepted Manuscript • DOI: 10.1021/la5017998 • Publication Date (Web): 12 Aug 2014 Downloaded from http://pubs.acs.org on August 12, 2014
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Spreading and Arrest of a Molten Liquid on Cold Substrates
F. Tavakoli1, Stephen H. Davis2, and H. P. Kavehpour1
1Complex
2Department
Fluids and Interfacial Physics Laboratory, UCLA, Department of Mechanical and Aerospace Engineering, Los Angeles, California 90024, USA
of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
Understanding spreading and solidification of liquids on cold solid surfaces is a problem of fundamental importance and general utility. The physics of non-isothermal spreading followed by phase change is still a mystery. The present work focuses on the dynamic and thermal characteristics of liquid drops spreading and their subsequent arrest due to freezing. Spreading of liquid is recorded and the evolution of liquid spread diameter and liquid-solid contact angle are measured from the recordings of a high-speed digital camera. After the initiation of solidification, the liquid drops are pinned to the substrate showing fixed footprints and contact angles. A physical hypothesis using scaling is provided to explain the relationship between the arrested base diameter ( D * ) and arrested contact angle ( θ * ) with respect to Stefan number (Ste). The experimental observations of solidified drops on cold substrates corroborate the derived physical theory. INTRODUCTION The non-isothermal spreading of liquid drops on smooth solid surfaces, though seemingly simple, are complicated free-boundary problems characterized by the presence of moving contact lines and temperature gradients. There are many instances of such dynamics in industrial applications such as thermal-spray coating1, rapid prototyping2, 3D printing3,4, plastic electronics5, welding6 and solder jetting in microelectronics7. It is also present in geophysics such as lava flows and gravity currents8,9. When a drop is gently deposited on a dry substrate, the spreading process is generally driven by interfacial forces, although inertial and gravitational forces can play significant roles. In
the present research, we focus exclusively on the system of drops on planar, cold solids at low Weber number, We = ρVR 2 σ , and high Ohnesorge number, Oh = µ
ρ Rσ , conditions where ρ, μ, V, σ
and R are the density, dynamic viscosity, mean velocity, surface tension and the drop radius, respectively. These parameters govern the spreading of the drop, whereas the Stefan number,
Ste = C p (Tm − T ) L f is the key factor in determining
the phase-change process and the subsequent arrest. Here C p , Tm , T and L f are the specific heat, the melting temperature of the liquid, the temperature of a solid substrate, and the latent
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heat of fusion of the liquid, respectively. The higher the Stefan number, the lower the substrate temperature. The first analytical study considering both simultaneous liquid flow and solidification10 gives a power-law relationship for the drop arrested base diameter as a function of Ste, with exponent 0.395. However, their experimental results do not fully match this behavior. Similarly, PasandidehFard et al.11 present a numerical solution using the continuum equations for momentum and energy for the modeling of the impact process. These results show agreement with the experimental data. A simpler model12, absent viscous forces, proposes that the solidification of a drop always occurs after droplet spreading is complete. The main questions arising in the physics of spreading and solidifying drops are: i) At what point on the solid substrate will the advancing triple line stop moving and, ii) What would be the arrested contact angle? In a series of articles, Sonin and co workers13-15 conduct low Weber deposition experiments looking into the effects of different parameters on the post-solidification shape of droplets. Small liquid droplets of mercury, water and wax are manually released with known frequency from a microliter syringe and allowed to fall onto a cold substrate of the same material. Prior to arrest, the relationship between the melt’s apparent contact angle and contact-line speed appears to obey Tanner-Hoffman’s law16,17:
(
)
U µ σ = K θ 3 − θ e3 where U is contact-line speed, θ
is apparent contact angle and θ e is equilibrium contact angle. The coefficient K depends on surface tension and viscosity. Sonin and co-workers13-15 assumed that the dynamic solidification front forms after solidification begins from the basal plane encompassing the trijunction. They contend that the moving drop stops spreading when the solidification contact angle approaches the
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apparent contact angle. When the solidification contact angle is near the value of the apparent contact angle, the drop spreading drastically decreases, and ultimately stops. For a given material, the relationship between the “spreading factor”, the dimensionless ratio, D* a, between the drop arrested base diameter and the drop’s initial diameter, and Ste is established empirically. The spreading factor is found to be inversely proportional to the one-third power of Ste, D* a ∝ Ste−1 3 , in the case of droplet deposition on like materials, provided that the contact angle is neither close to zero nor π14. This relationship is in accord with experimental results performed by Duthaler18. However, no physical justification is provided for this power-law relationship. From geometrical relations and a D* versus Ste equation, an implicit equation is derived for θ * in terms of Ste. Attinger et al.19 present the transient fluid dynamics and wetting of the novel solder-jetting technology. Their results suggest that solidification time depends non– monotonically on the substrate temperature. At lower Ste, the spread factor is practically independent of substrate temperature, whereas at high Ste, the spread factor decreases. It is asserted that contact-angle dynamics is strongly coupled with the evolution of the droplet free surface. For molten paraffin wax droplets on an aluminum surface, Bhola and Chandra20 observe that the arrested contact angle at a relatively low Ste equals the equilibrium contact angle, whereas a larger contact angle and smaller footprint is formed at higher Ste.
RESULTS AND DISCUSSION In this letter, we present a new analysis for the spreading and arrest of molten drops and new experiments of hexadecane, pentadecane and dodecane spreading on glass. We show that the
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model and experiments agree quantitatively and provide a practical description of the spreading behavior followed by arrest conditions.
this problem. In addition, the classical solidification model26 is used in the literature for the drop solidification problem10,27-29. We use the same model to find the growing length of the solid form, δ (t ) , starting from the contact line,
δ 2 ( t ) = 4 Ste α t
(1)
where t is time from the onset of solidification, and α is thermal diffusivity.
Figure 1. Schematics of the side view of the initial solidified volume of spreading drop on a cold solid substrate. Isotherms are shown as solid lines perpendicular to the liquid surface. The hatched region of volume Vc , shows an initial solidification at the trijunction. Drawing not to scale.
Here, we present a new hypothesis to describe the spreading-drop solidification. We assume that a drop stops spreading when the volume of solid formed at the trijunction, at which effectively infinite heat flux occurs, reaches a critical value, Vc (Fig. 1). The heat flux singularity stems from the discontinuous boundary condition of the heat flux vector at the trijunction15,21,22. This region emerges beneath an isotherm adjacent to the solid-liquidgas contact point. Because the air is a poor heat conductor compared to liquid, these isotherms are approximately perpendicular to the free surface of the liquid 23,24 and extend toward the fluid bulk; however, near the trijunction, isotherms extend thorough the substrate, but not very deep (not shown in Fig.1). It is also assumed that the average temperature of the substrate remains constant24,25. As the length, δ , of solidified layer near trijunction is much smaller than the drop diameter, one-dimensional analysis suffices for addressing
On the other hand, the volume of the drop at any given time, with a constant inflow, is
V ( t ) = Qt ≈ π D 2 h 4
(2)
where Q is the flow rate. For relatively small contact angles, h (drop height) can be approximated as Rθ , where R is the radius of the drop. The flow rate of liquid can also be scaled as,
Q≈
π D 3θ
(3)
8t
The critical volume of the solidified region, the hatched part in Fig. 1, can be estimated as,
θδ 2 Vc = π D 2
(4)
where Vc is the critical solidified volume, D = 2R is the diameter of a spreading drop. Equation 4 shows the volume of critical solidified region, which equals to the multiplication of drop circumference (πD) by the solidified circular sector at the trijunction (δθ2/2).
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We assume that t in equation 1 and 3 is measured from the onset of the solidification, so that we can eliminate t between these. Then, that gives a relationship between D* and the physical parameters. Substituting θ in equation 3 with Tanner-Hoffman’s law and replacing δ in equation 4 with equation 1 and combining resultant equations, the arrested base diameter can be related to Ste: D* −1 3 a = β Ste
β=
0.4Vc1 3Q1 6
(α 2 µ σ )
16
(5)
=
0.4Vc1 3Q1 6
( µ 3 ρ 2σ )
16
Pr1 3 .
(6)
where the Prandtl number is Pr = C p µ k and k is
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slides with dimensions of 50×24×0.15 mm3 (VWR microcover glass) are rinsed successively in acetone, methanol and deionized water. They are placed into a Krüss drop shape analyzer chamber, in which temperature and humidity are accurately controlled. The temperature of the glass substrate can be adjusted by a Peltier element ranging from 30 to 160 °C. The dispensing of liquid is conducted by the syringe-driver unit. The injection syringe situated on a syringe revolver is brought down automatically entering a hole from a top of the chamber close to the glass substrate. The distance between the injection needle and the glass substrate during liquid dispensing is kept constant and relatively high in the upper region of the drop so that the needle does not substantially alter the shape of the drop. Continuous dispensing of a liquid is performed with flow rates ranging from 0.2 to 600 µ L min.
thermal conductivity. We also note that D * Lcap = D*a aLcap . For our experiments, we use a series of alkane hydrocarbons on glass. Hexadecane, pentadecane and dodecane have similar physical properties except for the melting points, which directly enable us to investigate the influence of Ste on the postsolidification shape. These alkanes do not supercool and subsequently freeze just below melting temperature. The surface tensions are measured using the DuNoüy ring method (K100 tensiometer, Krüss) at room temperature. A drop-shape analysis system (DSA 100, Krüss) is used for recording the droplet spreading dynamics and evolution. This state-of-the-art apparatus provides accurate measurements of the dynamic contact angle and the drop base diameter of the spreading liquid. The intelligent dosing system of DSA 100 permits liquids to be dispensed without the risk of contaminating the sample. The bright illumination with extremely low radiated heat provides light that is required for measuring the drop’s evolution. In our experiments, glass
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Figure 2. Hexadecane spreading and solidification sequence on a cold glass substrate at a) 80ms, b) 1323ms, c) 1833ms, d) 3667ms after the touchdown (Ste = 0.03) with flow rate of 200 µl / min.
A high-speed camera is used to visualize the droplet spreading dynamics and phase change. The camera is capable of recording each event with a maximum of 99 frames per second. The drop base diameter and contact angle are measured at each frame by DSA 100 using a tangent method. In this method, the actual drop and the contact line with the solid are first determined by the analysis of the gray level values of the imaging pixels. From the fitted parameters, the slope of the three-phase contact point at the baseline is first determined and used to measure the contact angle. Consequently, the drop base diameter is defined as the distance between these two contact points. The contact angles mentioned are averages of the right and left contact angles measured by the DSA software. Visual observation acts as a direct indication of the start of drop solidification, as drops become instantly both immobile and opaque. Also, indirect methods such as sudden temporal independency of the base diameter data is interpreted as the solidification start.
* extent of spreading denoted by D Lcap , where the
capillary length (Fig. 3) is Lcap = σ ρ g , and g is the gravitational acceleration. Increasing the Ste number (colder substrate) causes the drop to solidify with a smaller footprint. Each data point represents six experiments. Variations of D* become larger at temperatures close to the melting points of hexadecane and pentadecane. At
Ste.Pr −1 3 = 0.2 , a drastic change in behavior is observed and the values of D* become more sensitive to Ste. At this point, θ * start to exceed a right angle and even reaches as high as 120°. Our power-law exponent of -1/3 follows the same power law as that of previous experiments conducted by Sonin’s group13,14,18. It should be noted that references 14 and 18 investigate the single drop deposition on cold substrates, whereas reference 13 studies the forced spreading of liquid on cold targets. Their solid substrates are of the same materials as the molten liquid, whereas ours use cover glass as a solid target for all our spreading and solidification tests.
Tests are run at various combinations of substrate temperatures and dispensing flow rates. After some amount of liquid spreading and subsequent solidification, arrested base diameter, * D * , and solidification contact angle, θ are defined at the moment that the drop gets pinned down due to solidification start (Fig. 2.b). Figure 2 shows a sequence of hexadecane spreading and subsequent arrest on a cold glass substrate (Ste = 0.03) with flow rate of 200 µl / min. It is worth mentioning
that the values of D * and θ * do not change in time after solidification inception (Fig. 2.d). The final post-solidification shape is characterized by the
Figure 3. Effect of substrate temperature on the arrested drop diameter. Dashed lines correspond to data on solidifying water, mercury, and solder drops14,18. Solid line represents our fitted equation ( D* Lcap = 0.58 ± 0.04 Ste−0.32±0.02 ) to our experimental data. Each data point is the average of 6 distinct experiments.
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The plot of arrested contact angle versus Ste is shown in Fig. 4. Each data point represents six experiments. Intuitively, with higher Ste, the drop solidifies more quickly leading to higher values of
θ * . Solidification contact angles for hexadecane and pentadecane ranges from 21° to 157.2°. θ * versus Ste plot also depicts limiting behavior: for Ste → ∞ (large supercooling), θ * → 180° (the drop
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Substituting Eq.7 into Eq. 5, gives an equation for solidification contact angle as a function of the Ste. Using Taylor series for the sine and cosine leads to14:
2 Ste 9 β3 1+ 19 3 − 1 . θ = 16 Ste β *
(8)
solidifies without apparent spreading), and for
Ste → 0 (small supercooling), θ * → 0 (the droplet spreads completely over the substrate). Our experimental results seem to diverge from Schiaffino’s13, because the molten liquid in Schiaffino’s experiments is deposited on a solid substrate of the same material causing local melting of the substrate at the vicinity of trijunction. This local melting could account for undervalued arrested contact angle measurements compared to actual contact angles. Assuming that the final solidified shape is a spherical cap, mass conservation relates θ * to D* a by:
(
)
8sin 1+ cosθ * D* = a 1− cosθ * 2 + cos θ *
(
)(
)
By putting the values of C obtained from the fitted power law equation to experimental data points of D * a versus Ste , modified function (8) fits our data starting from 0° and extending to 180°. The model seems to diverge from the experimental data at low Ste; however, the value of Ste = 0.0148 corresponds to the substrate temperature of 16.4 °C, very close to Tm = 18°C for hexadecane. It is possible that local temperature during the initial stage of the experiment is not very accurate due to thermal capacity of the substrate. In addition, because we measure the substrate temperature at the base of the substrate, temperature fluctuations of ±0.5 may cause such deviation from the theory.
(7)
For assuming that drops take a spherical cap form, Bond number should be less than unity. The Bond number is a measure of the importance of body forces compared to surface tension forces and is defined30 as Bo = ρ gH 2 σ where H is the drop height. Maximum values of Bond number for hexadecane, pentadecane and dodecane are 0.568, 0.754 and 0.652, respectively. The values of Bond number are less than unity and this indicates that we can neglect the role of hydrostatic pressure variations in distorting the shape of the drop and drops assume a spherical cap shape.
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Figure 4. Arrested contact angle versus Stefan number for hexadecane, pentadecane and dodecane on a glass target at different fluid flow rates ranging from 5 to 100 µl min . The black solid line represents our model fitted to 100 µl min hexadecane data for β = 0.7 . Dashed line corresponds to the results of molten wax on a solid wax from Schiaffino and Sonin13.
CONCLUSIONS In summary, we performed an experimental and theoretical study on the dynamics of molten drops on undercooled glass substrates, on which the drop spreads and comes to rest when it solidifies. Two sets of post-arrest geometrical parameters i) the arrested footprint ( D* ) and ii) the solidification contact angle ( θ * ) are experimentally obtained. These two values are measured using a high-speed camera situated in a drop-shape analyzer (DSA 100) machine. These results show a power-law relationship between the D* and the Ste for small arrested contact angles (
Letter
Spreading and Arrest of a Molten Liquid on Cold Substrates Faryar Tavakoli, Stephen Davis, and H. Pirouz Kavehpour Langmuir, Just Accepted Manuscript • DOI: 10.1021/la5017998 • Publication Date (Web): 12 Aug 2014 Downloaded from http://pubs.acs.org on August 12, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
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Spreading and Arrest of a Molten Liquid on Cold Substrates
F. Tavakoli1, Stephen H. Davis2, and H. P. Kavehpour1
1Complex
2Department
Fluids and Interfacial Physics Laboratory, UCLA, Department of Mechanical and Aerospace Engineering, Los Angeles, California 90024, USA
of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
Understanding spreading and solidification of liquids on cold solid surfaces is a problem of fundamental importance and general utility. The physics of non-isothermal spreading followed by phase change is still a mystery. The present work focuses on the dynamic and thermal characteristics of liquid drops spreading and their subsequent arrest due to freezing. Spreading of liquid is recorded and the evolution of liquid spread diameter and liquid-solid contact angle are measured from the recordings of a high-speed digital camera. After the initiation of solidification, the liquid drops are pinned to the substrate showing fixed footprints and contact angles. A physical hypothesis using scaling is provided to explain the relationship between the arrested base diameter ( D * ) and arrested contact angle ( θ * ) with respect to Stefan number (Ste). The experimental observations of solidified drops on cold substrates corroborate the derived physical theory. INTRODUCTION The non-isothermal spreading of liquid drops on smooth solid surfaces, though seemingly simple, are complicated free-boundary problems characterized by the presence of moving contact lines and temperature gradients. There are many instances of such dynamics in industrial applications such as thermal-spray coating1, rapid prototyping2, 3D printing3,4, plastic electronics5, welding6 and solder jetting in microelectronics7. It is also present in geophysics such as lava flows and gravity currents8,9. When a drop is gently deposited on a dry substrate, the spreading process is generally driven by interfacial forces, although inertial and gravitational forces can play significant roles. In
the present research, we focus exclusively on the system of drops on planar, cold solids at low Weber number, We = ρVR 2 σ , and high Ohnesorge number, Oh = µ
ρ Rσ , conditions where ρ, μ, V, σ
and R are the density, dynamic viscosity, mean velocity, surface tension and the drop radius, respectively. These parameters govern the spreading of the drop, whereas the Stefan number,
Ste = C p (Tm − T ) L f is the key factor in determining
the phase-change process and the subsequent arrest. Here C p , Tm , T and L f are the specific heat, the melting temperature of the liquid, the temperature of a solid substrate, and the latent
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heat of fusion of the liquid, respectively. The higher the Stefan number, the lower the substrate temperature. The first analytical study considering both simultaneous liquid flow and solidification10 gives a power-law relationship for the drop arrested base diameter as a function of Ste, with exponent 0.395. However, their experimental results do not fully match this behavior. Similarly, PasandidehFard et al.11 present a numerical solution using the continuum equations for momentum and energy for the modeling of the impact process. These results show agreement with the experimental data. A simpler model12, absent viscous forces, proposes that the solidification of a drop always occurs after droplet spreading is complete. The main questions arising in the physics of spreading and solidifying drops are: i) At what point on the solid substrate will the advancing triple line stop moving and, ii) What would be the arrested contact angle? In a series of articles, Sonin and co workers13-15 conduct low Weber deposition experiments looking into the effects of different parameters on the post-solidification shape of droplets. Small liquid droplets of mercury, water and wax are manually released with known frequency from a microliter syringe and allowed to fall onto a cold substrate of the same material. Prior to arrest, the relationship between the melt’s apparent contact angle and contact-line speed appears to obey Tanner-Hoffman’s law16,17:
(
)
U µ σ = K θ 3 − θ e3 where U is contact-line speed, θ
is apparent contact angle and θ e is equilibrium contact angle. The coefficient K depends on surface tension and viscosity. Sonin and co-workers13-15 assumed that the dynamic solidification front forms after solidification begins from the basal plane encompassing the trijunction. They contend that the moving drop stops spreading when the solidification contact angle approaches the
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apparent contact angle. When the solidification contact angle is near the value of the apparent contact angle, the drop spreading drastically decreases, and ultimately stops. For a given material, the relationship between the “spreading factor”, the dimensionless ratio, D* a, between the drop arrested base diameter and the drop’s initial diameter, and Ste is established empirically. The spreading factor is found to be inversely proportional to the one-third power of Ste, D* a ∝ Ste−1 3 , in the case of droplet deposition on like materials, provided that the contact angle is neither close to zero nor π14. This relationship is in accord with experimental results performed by Duthaler18. However, no physical justification is provided for this power-law relationship. From geometrical relations and a D* versus Ste equation, an implicit equation is derived for θ * in terms of Ste. Attinger et al.19 present the transient fluid dynamics and wetting of the novel solder-jetting technology. Their results suggest that solidification time depends non– monotonically on the substrate temperature. At lower Ste, the spread factor is practically independent of substrate temperature, whereas at high Ste, the spread factor decreases. It is asserted that contact-angle dynamics is strongly coupled with the evolution of the droplet free surface. For molten paraffin wax droplets on an aluminum surface, Bhola and Chandra20 observe that the arrested contact angle at a relatively low Ste equals the equilibrium contact angle, whereas a larger contact angle and smaller footprint is formed at higher Ste.
RESULTS AND DISCUSSION In this letter, we present a new analysis for the spreading and arrest of molten drops and new experiments of hexadecane, pentadecane and dodecane spreading on glass. We show that the
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model and experiments agree quantitatively and provide a practical description of the spreading behavior followed by arrest conditions.
this problem. In addition, the classical solidification model26 is used in the literature for the drop solidification problem10,27-29. We use the same model to find the growing length of the solid form, δ (t ) , starting from the contact line,
δ 2 ( t ) = 4 Ste α t
(1)
where t is time from the onset of solidification, and α is thermal diffusivity.
Figure 1. Schematics of the side view of the initial solidified volume of spreading drop on a cold solid substrate. Isotherms are shown as solid lines perpendicular to the liquid surface. The hatched region of volume Vc , shows an initial solidification at the trijunction. Drawing not to scale.
Here, we present a new hypothesis to describe the spreading-drop solidification. We assume that a drop stops spreading when the volume of solid formed at the trijunction, at which effectively infinite heat flux occurs, reaches a critical value, Vc (Fig. 1). The heat flux singularity stems from the discontinuous boundary condition of the heat flux vector at the trijunction15,21,22. This region emerges beneath an isotherm adjacent to the solid-liquidgas contact point. Because the air is a poor heat conductor compared to liquid, these isotherms are approximately perpendicular to the free surface of the liquid 23,24 and extend toward the fluid bulk; however, near the trijunction, isotherms extend thorough the substrate, but not very deep (not shown in Fig.1). It is also assumed that the average temperature of the substrate remains constant24,25. As the length, δ , of solidified layer near trijunction is much smaller than the drop diameter, one-dimensional analysis suffices for addressing
On the other hand, the volume of the drop at any given time, with a constant inflow, is
V ( t ) = Qt ≈ π D 2 h 4
(2)
where Q is the flow rate. For relatively small contact angles, h (drop height) can be approximated as Rθ , where R is the radius of the drop. The flow rate of liquid can also be scaled as,
Q≈
π D 3θ
(3)
8t
The critical volume of the solidified region, the hatched part in Fig. 1, can be estimated as,
θδ 2 Vc = π D 2
(4)
where Vc is the critical solidified volume, D = 2R is the diameter of a spreading drop. Equation 4 shows the volume of critical solidified region, which equals to the multiplication of drop circumference (πD) by the solidified circular sector at the trijunction (δθ2/2).
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We assume that t in equation 1 and 3 is measured from the onset of the solidification, so that we can eliminate t between these. Then, that gives a relationship between D* and the physical parameters. Substituting θ in equation 3 with Tanner-Hoffman’s law and replacing δ in equation 4 with equation 1 and combining resultant equations, the arrested base diameter can be related to Ste: D* −1 3 a = β Ste
β=
0.4Vc1 3Q1 6
(α 2 µ σ )
16
(5)
=
0.4Vc1 3Q1 6
( µ 3 ρ 2σ )
16
Pr1 3 .
(6)
where the Prandtl number is Pr = C p µ k and k is
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slides with dimensions of 50×24×0.15 mm3 (VWR microcover glass) are rinsed successively in acetone, methanol and deionized water. They are placed into a Krüss drop shape analyzer chamber, in which temperature and humidity are accurately controlled. The temperature of the glass substrate can be adjusted by a Peltier element ranging from 30 to 160 °C. The dispensing of liquid is conducted by the syringe-driver unit. The injection syringe situated on a syringe revolver is brought down automatically entering a hole from a top of the chamber close to the glass substrate. The distance between the injection needle and the glass substrate during liquid dispensing is kept constant and relatively high in the upper region of the drop so that the needle does not substantially alter the shape of the drop. Continuous dispensing of a liquid is performed with flow rates ranging from 0.2 to 600 µ L min.
thermal conductivity. We also note that D * Lcap = D*a aLcap . For our experiments, we use a series of alkane hydrocarbons on glass. Hexadecane, pentadecane and dodecane have similar physical properties except for the melting points, which directly enable us to investigate the influence of Ste on the postsolidification shape. These alkanes do not supercool and subsequently freeze just below melting temperature. The surface tensions are measured using the DuNoüy ring method (K100 tensiometer, Krüss) at room temperature. A drop-shape analysis system (DSA 100, Krüss) is used for recording the droplet spreading dynamics and evolution. This state-of-the-art apparatus provides accurate measurements of the dynamic contact angle and the drop base diameter of the spreading liquid. The intelligent dosing system of DSA 100 permits liquids to be dispensed without the risk of contaminating the sample. The bright illumination with extremely low radiated heat provides light that is required for measuring the drop’s evolution. In our experiments, glass
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Langmuir
Figure 2. Hexadecane spreading and solidification sequence on a cold glass substrate at a) 80ms, b) 1323ms, c) 1833ms, d) 3667ms after the touchdown (Ste = 0.03) with flow rate of 200 µl / min.
A high-speed camera is used to visualize the droplet spreading dynamics and phase change. The camera is capable of recording each event with a maximum of 99 frames per second. The drop base diameter and contact angle are measured at each frame by DSA 100 using a tangent method. In this method, the actual drop and the contact line with the solid are first determined by the analysis of the gray level values of the imaging pixels. From the fitted parameters, the slope of the three-phase contact point at the baseline is first determined and used to measure the contact angle. Consequently, the drop base diameter is defined as the distance between these two contact points. The contact angles mentioned are averages of the right and left contact angles measured by the DSA software. Visual observation acts as a direct indication of the start of drop solidification, as drops become instantly both immobile and opaque. Also, indirect methods such as sudden temporal independency of the base diameter data is interpreted as the solidification start.
* extent of spreading denoted by D Lcap , where the
capillary length (Fig. 3) is Lcap = σ ρ g , and g is the gravitational acceleration. Increasing the Ste number (colder substrate) causes the drop to solidify with a smaller footprint. Each data point represents six experiments. Variations of D* become larger at temperatures close to the melting points of hexadecane and pentadecane. At
Ste.Pr −1 3 = 0.2 , a drastic change in behavior is observed and the values of D* become more sensitive to Ste. At this point, θ * start to exceed a right angle and even reaches as high as 120°. Our power-law exponent of -1/3 follows the same power law as that of previous experiments conducted by Sonin’s group13,14,18. It should be noted that references 14 and 18 investigate the single drop deposition on cold substrates, whereas reference 13 studies the forced spreading of liquid on cold targets. Their solid substrates are of the same materials as the molten liquid, whereas ours use cover glass as a solid target for all our spreading and solidification tests.
Tests are run at various combinations of substrate temperatures and dispensing flow rates. After some amount of liquid spreading and subsequent solidification, arrested base diameter, * D * , and solidification contact angle, θ are defined at the moment that the drop gets pinned down due to solidification start (Fig. 2.b). Figure 2 shows a sequence of hexadecane spreading and subsequent arrest on a cold glass substrate (Ste = 0.03) with flow rate of 200 µl / min. It is worth mentioning
that the values of D * and θ * do not change in time after solidification inception (Fig. 2.d). The final post-solidification shape is characterized by the
Figure 3. Effect of substrate temperature on the arrested drop diameter. Dashed lines correspond to data on solidifying water, mercury, and solder drops14,18. Solid line represents our fitted equation ( D* Lcap = 0.58 ± 0.04 Ste−0.32±0.02 ) to our experimental data. Each data point is the average of 6 distinct experiments.
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The plot of arrested contact angle versus Ste is shown in Fig. 4. Each data point represents six experiments. Intuitively, with higher Ste, the drop solidifies more quickly leading to higher values of
θ * . Solidification contact angles for hexadecane and pentadecane ranges from 21° to 157.2°. θ * versus Ste plot also depicts limiting behavior: for Ste → ∞ (large supercooling), θ * → 180° (the drop
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Substituting Eq.7 into Eq. 5, gives an equation for solidification contact angle as a function of the Ste. Using Taylor series for the sine and cosine leads to14:
2 Ste 9 β3 1+ 19 3 − 1 . θ = 16 Ste β *
(8)
solidifies without apparent spreading), and for
Ste → 0 (small supercooling), θ * → 0 (the droplet spreads completely over the substrate). Our experimental results seem to diverge from Schiaffino’s13, because the molten liquid in Schiaffino’s experiments is deposited on a solid substrate of the same material causing local melting of the substrate at the vicinity of trijunction. This local melting could account for undervalued arrested contact angle measurements compared to actual contact angles. Assuming that the final solidified shape is a spherical cap, mass conservation relates θ * to D* a by:
(
)
8sin 1+ cosθ * D* = a 1− cosθ * 2 + cos θ *
(
)(
)
By putting the values of C obtained from the fitted power law equation to experimental data points of D * a versus Ste , modified function (8) fits our data starting from 0° and extending to 180°. The model seems to diverge from the experimental data at low Ste; however, the value of Ste = 0.0148 corresponds to the substrate temperature of 16.4 °C, very close to Tm = 18°C for hexadecane. It is possible that local temperature during the initial stage of the experiment is not very accurate due to thermal capacity of the substrate. In addition, because we measure the substrate temperature at the base of the substrate, temperature fluctuations of ±0.5 may cause such deviation from the theory.
(7)
For assuming that drops take a spherical cap form, Bond number should be less than unity. The Bond number is a measure of the importance of body forces compared to surface tension forces and is defined30 as Bo = ρ gH 2 σ where H is the drop height. Maximum values of Bond number for hexadecane, pentadecane and dodecane are 0.568, 0.754 and 0.652, respectively. The values of Bond number are less than unity and this indicates that we can neglect the role of hydrostatic pressure variations in distorting the shape of the drop and drops assume a spherical cap shape.
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Langmuir
Figure 4. Arrested contact angle versus Stefan number for hexadecane, pentadecane and dodecane on a glass target at different fluid flow rates ranging from 5 to 100 µl min . The black solid line represents our model fitted to 100 µl min hexadecane data for β = 0.7 . Dashed line corresponds to the results of molten wax on a solid wax from Schiaffino and Sonin13.
CONCLUSIONS In summary, we performed an experimental and theoretical study on the dynamics of molten drops on undercooled glass substrates, on which the drop spreads and comes to rest when it solidifies. Two sets of post-arrest geometrical parameters i) the arrested footprint ( D* ) and ii) the solidification contact angle ( θ * ) are experimentally obtained. These two values are measured using a high-speed camera situated in a drop-shape analyzer (DSA 100) machine. These results show a power-law relationship between the D* and the Ste for small arrested contact angles (
