An Overview Part III
Hewlett-Packard Labs Palo Alto November 13, 2007
Introduction to the Science & Technology of Coating and Drying Processes: An Overview Part III Brian G. Higgins Department of Chemical Engineering & Materials Science University of California, Davis
Outline of the talk • • • • • • • •
Begin with a brief overview of the Young-Laplace equation of interfacial statics, surface tension, contact angles, spreading and van der Waal forces Review of cellulose fibers, papermaking process, properties of paper, dimensional stability of paper to moisture. Review liquid coating operations, classification of coating methods, operating bounds for slot and roll coaters, its connection to capillary pressure. Use of computational methods to analyze, and probe a coating flow. Surface tension gradients- Marangoni effect. Example of a coating flow involving mass transfer (evaporation) and interfacial instability. Drying doing spin coating and the development of coating microstructure. Imbibition into porous substrates (paper), example of interface roughening, drops spreading on paper Methods for modeling flow in porous media: continuum approach versus network modeling; role of capillary pressure; some comments on drying of porous media
Kinetics of Liquid Penetration (Marmur & Cohen, 1997) Lucas-Washburn Equation.
Single horizontal capillary: Single vertical capillary:
Remarks: (i) The quantity
is linear in t no matter what A is.
(ii) One can write
and then
is independent of r !
Characterization of Porous Media (Marmur & Cohen, 1997) For a single capillary model, a plot of
is a universal curve! If the data do not follow the curve then one cannot assume porous medium can be characterized with a single equivalent pore radius
Single equivalent pore radius for Whatman filter paper
If multiple pore radii are required, one gets multiple curves
Which can be fit to data
Interface Roughening During Wetting Balankin et al (2000) Filter paper (thickness=0.25 mm, density=100 g/m2)
Chinese black ink (water and soot nanoparticles)
cf. Washburn Eqn:
Height of wetted area forms a Devil’s staircase. Paper structure gives rise to spasmodic flow. Mean height and interface width increases with time then saturate. Spasmodic flow evidence of lateral flow and/or unstable menisci -Haynes jump
Rough Interfaces with Precursor Wetting Low evaporation rates (high relative humidity) (Balankin et al, 2006)
Filter paper (thickness=0.25 mm, density=100 g/m2)
Chinese black ink (water and soot nanoparticles)
At high relative humidity (>40%), imbibition involves two simultaneous flows: film flow and bulk flow. Film flow propagates along the pore surfaces Bulk flow saturates the pore spaces.
Film flow front initially increases, but eventually bulk flow catches up.
Film flow and bulk flow mechanism can operate in paper wetting!
Some Definitions Permeability: the ability of a porous material to transmit a fluid Permeability (m2)
Darcy’s Law: Effective Permeability: The ability to preferentially flow a particular fluid through porous medium when other immiscible fluids are present. Effective Permeability (m2)
Relative Permeability: is the ratio of the effective permeability of a particular fluid (at a particular saturation S) to the absolute permeability of that fluid at total saturation. Relative Permeability
Saturation: is the ratio of the volume fraction of a particular fluid in a porous medium to the porosity of the porous medium
Multiphase Flow in Porous Media: Simplified Continuum Approach Volume fraction of phase= phase 1=rock, phase 2= oil(N), phase 3= water(W)
Porosity of porous media: Conservation of mass:
Fluid saturations:
Darcy’s Law:
Relative permeabilities: Capillary pressure:
Working Equations Requires consitutive relations that relate capillary pressure and relative permeability to saturation. Equations simplify when non-wetting phase is a gas.
Role of Capillary Pressure Consider a static blob of non-wetting liquid (N) trapped in a porous medium. Let the pressure in the blob be PN . What condition must be met in order for the wetting phase (W) to mobilize the blob?
Water flow direction
For non-wetting blob to squeeze through pore throat with radius the capillary pressure at the front of the blob must satisfy
Pressure drop in water phase over drop length L (Darcy’s Law):
Criteria to mobilize non-wetting blob
Capillary number
Thus the saturation level of the non-wetting phase will be a function of the capillary number
Spreading and Sorption of a Droplet on a Porous Substrate: Continuum Approach (Alleborn & Razzillier, 2004) Drop profiles and penetration depth
Model Details (i) (ii) (iii) (iv) (v) (vi) (vii)
Lubrication theory used for drop dynamics Evaporation effects ignored Darcy’s law for penetration into porous substrate Precursor films accounted for via disjoining pressure Discontinuous transition from saturated to unsaturated Various models were used to locate penetration front Governing equations solved by finite element method Drop profile Penetration front
Spreading and Sorption of a Droplet on a Porous Substrate: Continuum Approach (Alleborn & Razzillier, 2004)
Maximum drop radius Flat interface a consequence of ignoring lateral spreading
For t>0.5, R(t) decreases; contact line recedes.
Spreading and Imbibition of Liquid Droplets on Porous Surfaces (Clarke et al, 2002)
Drop volume as a function of time
Water/gycerol/hexylene gycol
Time increasing Millipore filters
Drop radius as a function of time
Model Details (i) (ii) (iii) (iv) (v) (vi)
Drop shape taken to be spherical during spreading Evaporation effects ignored Darcy’s law for penetration into porous substrate Molecular-kinetic theory for dynamic contact angle used Saturation front modeled using Washburn equation Governing equations solved numerically
Drops exhibit a maximum spreading radius then retracts
Network Model for Flow in Porous Media Mass balance over ith throat
flow direction
ith pore throat with radius Ri
Viscous pressure drop Between ith and jth throatt
periodic Balance Eq for ith throat:
fluid source flow conductance
Sum over all i and solve linear system to find pi’s
Flow rate through network: Determine Darcy permeability
Select radius from pore distribution function
Capillary Pressure Distribution in Pore Throat Positive capillary pressure gradient causes meniscus to jump to a stable Location- called the Haines jump
Gas
Liquid
Gas pressure need to force liquid through throat with radius
Meniscus position
Capillary pressure gradient positive Capillary pressure gradient negative
Drying of Porous Media Liquid evaporates
The basic mechanisms (L.E. Scriven et al, 1998) Menisci invade porous media
Menisci undergo Haines jumps as vapor diffuses out and drying continues
Vapor diffuses out via air-filled pore space
Continuum model not capable of capturing key mechanisms of drying and the influence of porous microstructure on menisci dynamics
Characterization of Paper Morphology (Ghassemzadeh & Sahimi, 2004)
SEM image of paper cross-section pore space fiber matrix epoxy resin
Throat size histogram fiber matrix pore space
Porosity, throat size, and connectivity conspire to determine permeability of a porous material to transmit a fluid Darcy’s Law:
Pore network simulation of fluid imbibition into paper during coating (Ghassemzadeh et al 2001,2004) Average pore throat diameter
Porous paper web
Model Details (i) Pore network for paper developed; takes into account anisotropic porosity, permeability. (ii) Lusas-Washburn kinetics used to advance interface In conjunction with Darcy flow in throats
Discontinuities in penetration depth arise because flow occurs in plane perpendicular to thickness direction
Introduction to the Science & Technology of Coating and Drying Processes: An Overview Part III Brian G. Higgins Department of Chemical Engineering & Materials Science University of California, Davis
Outline of the talk • • • • • • • •
Begin with a brief overview of the Young-Laplace equation of interfacial statics, surface tension, contact angles, spreading and van der Waal forces Review of cellulose fibers, papermaking process, properties of paper, dimensional stability of paper to moisture. Review liquid coating operations, classification of coating methods, operating bounds for slot and roll coaters, its connection to capillary pressure. Use of computational methods to analyze, and probe a coating flow. Surface tension gradients- Marangoni effect. Example of a coating flow involving mass transfer (evaporation) and interfacial instability. Drying doing spin coating and the development of coating microstructure. Imbibition into porous substrates (paper), example of interface roughening, drops spreading on paper Methods for modeling flow in porous media: continuum approach versus network modeling; role of capillary pressure; some comments on drying of porous media
Kinetics of Liquid Penetration (Marmur & Cohen, 1997) Lucas-Washburn Equation.
Single horizontal capillary: Single vertical capillary:
Remarks: (i) The quantity
is linear in t no matter what A is.
(ii) One can write
and then
is independent of r !
Characterization of Porous Media (Marmur & Cohen, 1997) For a single capillary model, a plot of
is a universal curve! If the data do not follow the curve then one cannot assume porous medium can be characterized with a single equivalent pore radius
Single equivalent pore radius for Whatman filter paper
If multiple pore radii are required, one gets multiple curves
Which can be fit to data
Interface Roughening During Wetting Balankin et al (2000) Filter paper (thickness=0.25 mm, density=100 g/m2)
Chinese black ink (water and soot nanoparticles)
cf. Washburn Eqn:
Height of wetted area forms a Devil’s staircase. Paper structure gives rise to spasmodic flow. Mean height and interface width increases with time then saturate. Spasmodic flow evidence of lateral flow and/or unstable menisci -Haynes jump
Rough Interfaces with Precursor Wetting Low evaporation rates (high relative humidity) (Balankin et al, 2006)
Filter paper (thickness=0.25 mm, density=100 g/m2)
Chinese black ink (water and soot nanoparticles)
At high relative humidity (>40%), imbibition involves two simultaneous flows: film flow and bulk flow. Film flow propagates along the pore surfaces Bulk flow saturates the pore spaces.
Film flow front initially increases, but eventually bulk flow catches up.
Film flow and bulk flow mechanism can operate in paper wetting!
Some Definitions Permeability: the ability of a porous material to transmit a fluid Permeability (m2)
Darcy’s Law: Effective Permeability: The ability to preferentially flow a particular fluid through porous medium when other immiscible fluids are present. Effective Permeability (m2)
Relative Permeability: is the ratio of the effective permeability of a particular fluid (at a particular saturation S) to the absolute permeability of that fluid at total saturation. Relative Permeability
Saturation: is the ratio of the volume fraction of a particular fluid in a porous medium to the porosity of the porous medium
Multiphase Flow in Porous Media: Simplified Continuum Approach Volume fraction of phase= phase 1=rock, phase 2= oil(N), phase 3= water(W)
Porosity of porous media: Conservation of mass:
Fluid saturations:
Darcy’s Law:
Relative permeabilities: Capillary pressure:
Working Equations Requires consitutive relations that relate capillary pressure and relative permeability to saturation. Equations simplify when non-wetting phase is a gas.
Role of Capillary Pressure Consider a static blob of non-wetting liquid (N) trapped in a porous medium. Let the pressure in the blob be PN . What condition must be met in order for the wetting phase (W) to mobilize the blob?
Water flow direction
For non-wetting blob to squeeze through pore throat with radius the capillary pressure at the front of the blob must satisfy
Pressure drop in water phase over drop length L (Darcy’s Law):
Criteria to mobilize non-wetting blob
Capillary number
Thus the saturation level of the non-wetting phase will be a function of the capillary number
Spreading and Sorption of a Droplet on a Porous Substrate: Continuum Approach (Alleborn & Razzillier, 2004) Drop profiles and penetration depth
Model Details (i) (ii) (iii) (iv) (v) (vi) (vii)
Lubrication theory used for drop dynamics Evaporation effects ignored Darcy’s law for penetration into porous substrate Precursor films accounted for via disjoining pressure Discontinuous transition from saturated to unsaturated Various models were used to locate penetration front Governing equations solved by finite element method Drop profile Penetration front
Spreading and Sorption of a Droplet on a Porous Substrate: Continuum Approach (Alleborn & Razzillier, 2004)
Maximum drop radius Flat interface a consequence of ignoring lateral spreading
For t>0.5, R(t) decreases; contact line recedes.
Spreading and Imbibition of Liquid Droplets on Porous Surfaces (Clarke et al, 2002)
Drop volume as a function of time
Water/gycerol/hexylene gycol
Time increasing Millipore filters
Drop radius as a function of time
Model Details (i) (ii) (iii) (iv) (v) (vi)
Drop shape taken to be spherical during spreading Evaporation effects ignored Darcy’s law for penetration into porous substrate Molecular-kinetic theory for dynamic contact angle used Saturation front modeled using Washburn equation Governing equations solved numerically
Drops exhibit a maximum spreading radius then retracts
Network Model for Flow in Porous Media Mass balance over ith throat
flow direction
ith pore throat with radius Ri
Viscous pressure drop Between ith and jth throatt
periodic Balance Eq for ith throat:
fluid source flow conductance
Sum over all i and solve linear system to find pi’s
Flow rate through network: Determine Darcy permeability
Select radius from pore distribution function
Capillary Pressure Distribution in Pore Throat Positive capillary pressure gradient causes meniscus to jump to a stable Location- called the Haines jump
Gas
Liquid
Gas pressure need to force liquid through throat with radius
Meniscus position
Capillary pressure gradient positive Capillary pressure gradient negative
Drying of Porous Media Liquid evaporates
The basic mechanisms (L.E. Scriven et al, 1998) Menisci invade porous media
Menisci undergo Haines jumps as vapor diffuses out and drying continues
Vapor diffuses out via air-filled pore space
Continuum model not capable of capturing key mechanisms of drying and the influence of porous microstructure on menisci dynamics
Characterization of Paper Morphology (Ghassemzadeh & Sahimi, 2004)
SEM image of paper cross-section pore space fiber matrix epoxy resin
Throat size histogram fiber matrix pore space
Porosity, throat size, and connectivity conspire to determine permeability of a porous material to transmit a fluid Darcy’s Law:
Pore network simulation of fluid imbibition into paper during coating (Ghassemzadeh et al 2001,2004) Average pore throat diameter
Porous paper web
Model Details (i) Pore network for paper developed; takes into account anisotropic porosity, permeability. (ii) Lusas-Washburn kinetics used to advance interface In conjunction with Darcy flow in throats
Discontinuities in penetration depth arise because flow occurs in plane perpendicular to thickness direction
