Derivation of Equations

8/2/2019 Derivation of Equations

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Derivation of Equations

Article Langmuir 2010, 26(12),

10365-10372


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Assumption Spherical droplets of radius R0 (Dia.-D0) takes the form of spherical cap of radius Rf; Possible effects caused by temperature variations are neglected;

In addition, we assume that the time scale for evaporation of the droplet is much larger than the time scale of the impactprocess;

After impact the fluid flow velocity is zero and the droplet has taken the shape of a spherical cap with contact angle

Weber Number:

Reynold Number:

1(a)

1(b)


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Energy Balance Equation

whereRf is the final radius after impact has completed, fS is the ratio of the fluidvapor

surface and the fluidsolid surface, and Eu is the dissipated energy in the impact process

by viscosity. ls and sv are the surface energies of the liquidsolid and solidvapor

interfaces, respectively.

(2)


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Youngs Energy Balance

sv ls = cos() (3)

Volume Conservation:From Geometry--

Volume of Sphere Volume of Spherical Cap

(4)


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Using (1a), (2), (3) and (4) and neglecting Eu,

the following is obtained:

(5)

From Equation (4),

(6)


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Drop-Track Mass Conservation

L (7)

r radius of the cylindrical

cap (analogous to Rf)


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From (7),

Where p drop pitch and equals to L/N

Maximum value of when Rf= Ro

r is analogous to Rf; w* is the dimensionless width

Maximum Pitch (pmax at Rf= R0) is given by

pmax=

(8)

(9a)


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In dimensionless form,

(9b)


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Dynamic Line Stability- A further bulging instability is observed at smaller droplet

separation, which consists of a series of bulges connected by ridges of width Rf.

According to Duineveld, transition to this morphology can be explained by an axial

flow within the bead that occur over a timescale significantly shorter than for

capillary spreading. He said that there is a flow, Q, along the bead driven by a

pressure difference from any variation in bead width. If Q>>QA, where QA is the

applied flow caused by the deposition of droplets, then any newly deposited liquid

will preferentially flow along the bead to the area of low pressure rather than causing

the bead to spread.


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Generalization of Dynamic Line Stability

According to Duineveld, QA is the mean flow rate, generated by droplet

deposition, is significantly slower than the axial flow rate, Q, due to pressuredifferences within the bead, i.e.,

(10)

or, (11)

where k1 is a constant and >1;

QA could be defined by the rate of arrival of drops from the printer,

where VD is the volume of a droplet, f is the frequency of droplet generation,

and UT is the speed at which the substrate moves relative to the print head.

(12)


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According to Berger, Volumetric rate of flow through ridge, Q, is

given by:

(13)

where PR Pressure difference b/w ridge and bulge;

AR Cross-sectional area of the ridge;

Dynamic viscosity of the liquid;lr length of the ridge between the droplets;

and S shape factor (a function of the contact angle, ..

According to Duineveld,

Now, the pressure difference PR b/w ridge & droplet could be

generalized as

(14)


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Now, approximately,

lr =p (15)

Also, from mass conservation within the bead, the area of cross-section of

ridge is given by:

From Equations (11) to (16),

(17)

In dimensionless drop spacing, p*, equation (17) could be expressed as:

(16)

(18)

Derivation of Equations

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