CN108804760A - A kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis of horizontal rule - Google Patents
A kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis of horizontal rule Download PDFInfo
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Abstract
A kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis of horizontal rule, including：Drop profile discretization；The derivation of the system dimensionless free energy function of four kinds of different wetting contact conditions；The wetting state for calculating heavy-fluid drop when contact circle is fixed, contacts radius of circle and the penetration depth in level-one (two level) mechanism；Initial volume increment is set, corresponding minimum dimensionless free energy is calculated under the conditions of contact radius of circle is fixed；When contact circle will change, current Local Minimum dimensionless free energy is calculated；By using binary system iterative method, may search for a dilatation and surface contact angle (advancing angle) so that current minimum dimensionless free energy and Local Minimum dimensionless free energy are only poor to be equal to energy barrier；Setting volume is initially reduced.Researcher can be helped to further appreciate that the dynamic property of drop, there is great importance for the design of microfluidic system.
Description
Technical field
The invention belongs to micro-nano compound structure technical fields, are related to a kind of double roughness surface heavy-fluid drop contacts of horizontal rule
The computational methods of angular lag, and in particular to a kind of that level indicator being calculated using minimum free energy theory and nonlinear optimization algorithm
The then contact angle hysteresis (CAH) of double roughness surfaces heavy-fluid drop.
Background technology
For the heavy-fluid drop balanced on rough surface, when interval Vr is reduced or increased in droplet size V by us<V<Va,
But when three-phase line of contact is constant, the liquid-drop contact angle corresponding to Vr and Va is just receding contact angle and advancing contact angle respectively.And
Contact angle hysteresis is represented by：CAH=θɑ-θr.Contact angle hysteresis (CAH) is an important performance parameter of heavy-fluid drop, is used for
The mobile property of drop on the surface is described.CAH is smaller, drop easier movement on the surface.Accordingly, it is determined that rough surface
The contact angle hysteresis (CAH) of the heavy-fluid drop of upper balance, the dynamic property for studying drop have great importance, help to manage
Solution heavy-fluid drops in wetting and diffusion property on functional hierarchy surface, simultaneously, moreover it can be used to instruct the design of microfluidic system, have
There is important realistic meaning.
Currently, not having a kind of specific computational methods also to calculate contact angle hysteresis (CAH).Therefore, in order to thick
The contact angle hysteresis (CAH) of drop, research and the dynamic characteristic for weighing rough surface drop are fast and accurately obtained on rough surface,
There is an urgent need to a kind of simple, perfect theoretical analysis methods, calculate the contact angle hysteresis (CAH) for obtaining drop.
Invention content
The present invention will overcome the disadvantages mentioned above of the prior art, provide a kind of double roughness surface heavy-fluid drop contacts of horizontal rule
The computational methods of angular lag (CAH) can calculate the contact angle hysteresis (CAH) that heavy-fluid is dripped on rough surface.
The present invention devises a computational methods, can be obtained on the double roughness surfaces of horizontal rule using this method
Heavy-fluid drips contact angle hysteresis (CAH).For the double roughness surfaces of horizontal rule, only it is to be understood that its liquid gas interface coefficient of tension, liquid
Drop in the equivalent contact angle in primary structure, secondary structure and substrate, droplet size, drop density, substrate, primary structure and
Energy barrier in secondary structure and structure size pass through calculating, it will be able to which theoretically drop contact is calculated in analysis
Angular lag (CAH) instructs research drop the design of microfluidic system in the dynamic property and practical application of rough surface
With great importance.
A kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis (CAH) of horizontal rule, include the following steps：
(1) assume a model by drop and micro-nano secondary structure surface composition, wherein first order nanostructure and second level micron
Structure is all square column, and geometric scale periodic distribution, secondary structure and substrate are made of different materials.Total apparent face of model
Product is length of side L0Square, the length of side be much larger than liquid-drop diameter.a1, b1, h1And x1Be respectively the first level structure square column the length of side,
Away from, height and invasive depth, a2, b2, h2And x2It is the length of side, spacing, height and the invasive depth of the second level structure square column respectively,
The size of first level structure is much smaller than the second level structure.Since the heavy droplet profile on double roughness surfaces of horizontal rule is axis
Symmetrically, so considering this problem on two dimensional surface.It is v axis to select the symmetry axis that heavy-fluid is dripped, and the surface of heavy-fluid drop connects
Contacting surface is u axis, and the contour curve that heavy-fluid is dripped is divided into N equal portions along the directions v using finite difference calculus, obtains discrete point：
Wherein, haIt is the maximum height of drop, uiIt is the abscissa of discrete point, viIt is the ordinate of discrete point.
The derivation of the system dimensionless free energy function of (2) four kinds of different wetting contact conditions, derivation are as follows：
21. dropping in the penetration depth in primary structure and secondary structure by heavy-fluid, wetting contact state is divided into four kinds,
Respectively W-W, C-W, W-C and C-C state, partitioning standards are as follows：
(a) primary structure and secondary structure are in Wenzel states, i.e. x1=h1, x2=h2, this contact condition name
For W-W states.
(b) primary structure is in Cassie states or contact condition will be changed into Wenzel states from Cassie states,
Secondary structure is in Wenzel states, i.e. 0≤x1<h1, x2=h2, this contact condition is named as C-W states.
(c) primary structure is in Wenzel states, and secondary structure is in Cassie states or contact condition will be from
Cassie states are changed into Wenzel states, i.e. x1=h1, 0≤x2<h2, this contact condition is named as W-C states.
(d) primary structure and secondary structure are in Cassie states or contact condition and will be changed into from Cassie states
Wenzel states, i.e. 0≤x1<h1, 0≤x2<h2, this contact condition is named as C-C states.
Wherein, x1It is penetration depth of the drop in primary structure, x2It is penetration depth of the drop in secondary structure, h1
It is the height of primary structure, h2It is the height of secondary structure.
22. the system free energy E of four kinds of different wetting contact conditionsIEqual to EI=Ea+Eb, I=1,2,3,4.Subscript I=
1,2,3,4 indicate W-W, C-W, W-C and C-C state respectively.EaAnd EbRespectively represent gravitional force and surface free energy.
EaIt can be expressed as：
EbIt can be expressed as：
Eb=γLVSLV+γSL0SSL0+γSL1SSL1+γSL2SSL2+γSV0SSV0+γSV1SSV1+γSV2SSV2 (3)
Wherein, ρ is fluid density, and g is acceleration of gravity, and S is the contact area at interface, and γ is interfacial tension coefficient, under
Mark LV, SL and SV respectively represent liquid-gas, solid-liquid and solid-air interface, and subscript 1,2 and 0 respectively represents primary structure, secondary structure
And substrate.
The external surface area S of dropextIt can be expressed as：
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop.
Drop and the contact surface area of substrate can be expressed as：
Sbase=π rb 2 (5)
In formula, rbRadius of circle is contacted for drop and two level micro-nano structure surface.
According to the difference of each wet face state, can be easy to acquire the S under four kinds of statesLV, SSV0, SSV1, SSV2, SSL0,
SSL1, SSL2,Corresponding expression formula:
(a) W-W states:
SLV=Sext=2 π R2(1-cosθ) (6)
SSL0=Sbase(1-f1)(1-f2)=π rb 2(1-f1)(1-f2) (7)
SSL1=Sbaser2[r1-(1-f1)]=π rb 2r2(r1-1+f1) (8)
SSL2=Sbase(1-f1)[r2-(1-f2)]=π rb 2(1-f1)(r2-1+f2) (9)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(b) C-W states
SSL0=0 (14)
SSL2=0 (16)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(c) W-C states：
SSL0=0 (21)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(d) C-C states：
SSL0=0 (28)
SSL2=0 (30)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
It brings parameter into formula and can obtain the system free energys of four kinds of different wetting contact conditions and be：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, f1It is level-one knot
The solid portion of structure, f2It is the solid portion of secondary structure, is solid area and surface area ratio, r in contact surface1It is level-one knot
The roughness value of structure, r2The roughness value of secondary structure, i.e., the practical gross area and surface area ratio, ɑ1And ɑ2It is respectively
Each column array section length of side, r in level-one, secondary structurebRadius of circle is contacted for drop and two level micro-nano structure surface；x1To receive
Invasive depth under rice structure, x2For the invasive depth under micrometer structure.
23. the system dimensionless free energy E of four kinds of different wetting contact conditionsI' determination, I=1,2,3,4 respectively represent
W-W, C-W, W-C and C-C state：When given heavy-fluid drop and the physical property of double rough surfaces,It is constant, and by Young's equation
Equivalent contact angle, which can be obtained, isThe system free energy of four kinds of different wetting contact conditions can be reduced to：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
In formula, θe1, θe2And θe0Indicate that flat surfaces upper liquid drops in primary structure respectively, the sheet in secondary structure and substrate
Levy contact angle；f1It is the solid portion of primary structure, f2It is the solid portion of secondary structure, is solid area and table in contact surface
Area ratio.r1It is the roughness value of primary structure, r2It is the i.e. practical gross area of roughness value and surface area of secondary structure
The ratio between；ɑ1And ɑ2It is each column array section length of side, r in level-one, secondary structure respectivelybFor drop and two level micro-nano structure surface
Contact radius of circle；x1For the invasive depth under nanostructure, x2For the invasive depth under micrometer structure.
(3) the wetting state I of current heavy-fluid drop is calculatedf, contact radius of circle rb-f, infiltration of the drop in primary structure
Depth x1-fWith penetration depth x of the drop in grade level structure2-f
31. the optimization of Local Minimum dimensionless free energy under four kinds of contact conditions：
311. variable：Discrete point coordinates ui, vi, i=1,2 ..., N+1, drop is in primary structure x1With secondary structure x2In
Penetration depth；
312. constraint:
i.
ii.
iii.0≤x1< h1,0≤x2< h2 (45)
Iv. because contour curve function u=f (v) is a convex function, so
v.
Wherein, hɑmaxIt is the maximum height of drop, V is the volume of drop, uiIt is the abscissa of discrete point, viIt is discrete point
Ordinate, rbmaxIt is the maximum value of contact line radius, x1It is penetration depth of the drop in primary structure, x2It is drop two
Penetration depth in level structure, h1It is the height of primary structure, h2It is the height of secondary structure；
313. search:
The search variables x under constraints1,x2,h1,h2, best object is obtained simultaneously using Matlab functions " fmincon "
Calculate the part minimum dimensionless free energy E under four kinds of wetting contact statesImin', I=1,2,3,4；
32. the optimization of whole minimum dimensionless free energy under four kinds of contact conditions：Using Matlab functions " fmincon " come
The minimum dimensionless free energy E in search partImin', I=1,2,3,4, in whole minimum dimensionless free energy Emin', constantly count
It calculates, until whole minimum dimensionless free energy difference converges to 10-4Tolerance interval in；At this moment dimensionless free energy is considered as
It is whole minimum dimensionless free energy, the wetting state I of current heavy-fluid drop under meterf, contact radius of circle rb-f, drop is in level-one knot
Penetration depth x in structure1-fWith penetration depth x of the drop in grade level structure2-f；Otherwise, isodisperse N is doubled, is walked again
Rapid 31 and step 32.
(4) setting initial volume increment is Δ V=0.5V, and contact radius of circle is fixed and u1=rb=rb-f, x1=x1-f, meter
Calculate corresponding minimum dimensionless free energy EV+ΔV-fix', i.e. EImin'.If u1=rb, v1=0, because of I, rb,x1And x2For definite value when,
So minimum dimensionless free energy is optimized for：
(a) variable:Discrete point coordinates ui, vi,
(b) setting optimization object EI'minWith constraints i, ii, iii, iv and v, with step 312.
(c) the search variables u under constraintsi, vi, i=2 ... .., N+1 are utilized under the conditions of contact circle is fixed
Fmincon functions in Matlab obtain optimal object and calculate whole minimum dimensionless free energy EImin'.Constantly counted
It calculates, until whole minimum dimensionless free energy difference converges to 10-4Tolerance interval in, dimensionless free energy at this moment is considered as
It is whole minimum dimensionless free energy EImin', i.e., the corresponding minimum dimensionless free energy of heavy-fluid drop that volume is Δ V+V
EV+ΔV-fix'.Write down discrete point p simultaneouslyi-f[u0,v0] and pi-f[ui,vi] and obtain corresponding contour curve and corresponding surface
Contact angle passes through formulaOtherwise, isodisperse N is just doubled, repeats step (b), (c).
(5) when contact radius of circle change when, i.e., contact circle variation when, Local Minimum dimensionless free energy from become EV+ΔV-fix’
For EV+ΔV', step (3) is repeated, E is calculatedV+ΔV'=Emin’。
(6) CAH be as contact circle on energy barrier caused by, can be by the way that E be calculatedbarr=U0l0+U1l1+U2l2,
In formula, EbarrIt is energy barrier, U0, U1And U2It is the energy barrier between drop and substrate, primary structure and secondary structure respectively；
l0, l1, l2It is the physical length between drop and substrate, primary structure and secondary structure respectively.When contact circle variation, part is most
Small dimensionless free energy is from EV+ΔV-fix' become EV+ΔV'.Δ E=EV+ΔV-fix’-EV+ΔV'>0, it is found that when contact circle variation, connects
Tactile circle obtains potential energy Δ E.As Δ E=EbarrWhen, contacting circle can forward, this ACA for being is exactly θɑ。
(7) by using binary system iterative method, Δ V and corresponding θ be may search forɑ=θV+ΔV-fixMake EV+ΔV-fix’-
EV+ΔV'=Ebarr。
(8) setting initial volume increment is Δ V=-0.5V and calculates θ using identical methodr, and calculate contact
Angular lag CAH=θɑ-θr。
The present invention inventive point be：Propose a kind of calculating calculating heavy-fluid drop contact angle hysteresis (CAH) on rough surface
Method, only it is to be understood that its liquid gas interface coefficient of tension, equivalent contact angle of the drop in primary structure, secondary structure and substrate,
Droplet size, drop density, the energy barrier in substrate, primary structure and secondary structure and structure size, by calculating,
It can theoretically analyze and drop contact angular lag (CAH) is calculated, researcher is helped to further understand rough surface
Dynamic and rough surface wetability, for instructing the design of microfluidic system to anticipate with important guiding in practical application
Justice.
Description of the drawings
Fig. 1 a are the side schematic views of the system model of the embodiment of the present invention；
Fig. 1 b are the three dimension system model schematics of the embodiment of the present invention；
Fig. 1 c are the sketches of the firsts and seconds structure of the embodiment of the present invention；
Fig. 2 a~Fig. 2 d are the wetting schematic diagrames that heavy-fluid of the embodiment of the present invention drops in secondary structure surface, wherein Fig. 2 a generations
Table W-W states, Fig. 2 b represent C-W states, and Fig. 2 c represent W-C states, and Fig. 2 d represent C-C states.
Fig. 3 is the half-sectional view of heavy-fluid of embodiment of the present invention drop；
Fig. 4 a are the schematic diagrames of the variation of dimensionless free energy and ACA after heavy-fluid drop volume becomes larger；
Fig. 4 b are the schematic diagrames of the variation of dimensionless free energy and ACA after heavy-fluid drop volume becomes smaller；
Fig. 5 is variation diagrams of the CAH with Bond numbers (Bo).
Specific implementation mode
The invention will be further described below in conjunction with the accompanying drawings and by specific embodiment, and following embodiment is descriptive
, it is not restrictive, protection scope of the present invention cannot be limited with this.
A kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis of horizontal rule, include the following steps：
(1) we consider a regular double roughness surfaces, primary structure and secondary structure are all square columns, and
And their geometric dimension is all periodic.As shown in Figure 1, the system is by a drop and a rectangular shaped surface area group
At, and the length of side L of rectangular shaped surface area0It is constant and be more than liquid-drop diameter.a1, b1, h1And x1It is the first level structure square column respectively
The length of side, spacing, height and invasive depth, a2, b2, h2And x2Be respectively the length of side of the second level structure square column, spacing, height and
The size of invasive depth, the first level structure is much smaller than the second level structure, outer interface area and surface area contact.f1It is level-one
The solid portion of structure, f2The solid portion of secondary structure, i.e., solid area and surface area ratio in contact surface,r1It is the roughness value of primary structure, r2It is the roughness value of secondary structure,
The i.e. practical gross area and surface area ratio,Due to double roughness of horizontal rule
Heavy droplet profile on surface is axisymmetric, so considering this problem on two dimensional surface.As shown in figure 3, selection weight
The symmetry axis of drop is v axis, and the surface contact surface of heavy-fluid drop is u axis, and the contour curve for utilizing finite difference calculus to drip heavy-fluid
It is divided into N equal portions along the directions v, obtains discrete point：
Wherein, haIt is the maximum height of drop, uiIt is the abscissa of discrete point, viIt is the ordinate of discrete point.
The derivation of the system dimensionless free energy function of (2) four kinds of different wetting contact conditions, derivation are as follows：
21. dropping in the penetration depth in primary structure and secondary structure by heavy-fluid, wetting contact state is divided into four kinds,
Respectively W-W, C-W, W-C and C-C state, partitioning standards are as follows：
(a) primary structure and secondary structure are in Wenzel states, i.e. x1=h1, x2=h2, this contact condition name
For W-W states.
(b) primary structure is in Cassie states or contact condition will be changed into Wenzel states from Cassie states,
Secondary structure is in Wenzel states, i.e. 0≤x1<h1, x2=h2, this contact condition is named as C-W states.
(c) primary structure is in Wenzel states, and secondary structure is in Cassie states or contact condition will be from
Cassie states are changed into Wenzel states, i.e. x1=h1, 0≤x2<h2, this contact condition is named as W-C states.
(d) primary structure and secondary structure are in Cassie states or contact condition and will be changed into from Cassie states
Wenzel states, i.e. 0≤x1<h1, 0≤x2<h2, this contact condition is named as C-C states.
Wherein, x1It is penetration depth of the drop in primary structure, x2It is penetration depth of the drop in secondary structure, h1
It is the height of primary structure, h2It is the height of secondary structure.
22. the system free energy E of four kinds of different wetting contact conditionsIEqual to EI=Ea+Eb, I=1,2,3,4.Subscript I=
1,2,3,4 indicate W-W, C-W, W-C and C-C state respectively.Ea and EbRespectively represent gravitional force and surface free energy.
Ea can be expressed as：
EbIt can be expressed as：
Eb=γLVSLV+γSL0SSL0+γSL1SSL1+γSL2SSL2+γSV0SSV0+γSV1SSV1+γSV2SSV2 (3)
Wherein, ρ is fluid density, and g is acceleration of gravity, and S is the contact area at interface, and γ is interfacial tension coefficient, under
Mark LV, SL and SV respectively represent liquid-gas, solid-liquid and solid-air interface, and subscript 1,2 and 0 respectively represents primary structure, secondary structure
And substrate.
The external surface area S of dropextIt can be expressed as：
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop.
Drop and the contact surface area of substrate can be expressed as：
Sbase=π rb 2(5)
In formula, rbRadius of circle is contacted for drop and two level micro-nano structure surface.
According to the difference of each wet face state, can be easy to acquire the S under four kinds of statesLV, SSV0, SSV1, SSV2, SSL0,
SSL1, SSL2, corresponding expression formula:
(a) W-W states:
SLV=Sext=2 π R2(1-cosθ) (6)
SSL0=Sbase(1-f1)(1-f2)=π rb 2(1-f1)(1-f2) (7)
SSL1=Sbaser2[r1-(1-f1)]=π rb 2r2(r1-1+f1) (8)
SSL2=Sbase(1-f1)[r2-(1-f2)]=π rb 2(1-f1)(r2-1+f2) (9)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(b) C-W states
SSL0=0 (14)
SSL2=0 (16)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(c) W-C states：
SSL0=0 (21)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
(d) C-C states：
SSL0=0 (28)
SSL2=0 (30)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
It brings parameter into formula and can obtain the system free energys of four kinds of different wetting contact conditions and be：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown
The external surface area of drop, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micron knot
The roughness factor of structure, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For rectangular shaped surface area
The length of side, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micron
Invasive depth under structure.
23. the system dimensionless free energy E of four kinds of different wetting contact conditionsI' determination, I=1,2,3,4 respectively represent
W-W, C-W, W-C and C-C state：When given heavy-fluid drop and the physical property of double rough surfaces,
It is constant, and by
Young's equation can obtain equivalent contact angleThe system free energy of four kinds of different wetting contact conditions can letter
It turns to：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, f1It is level-one knot
The solid portion of structure, f2It is the solid portion of secondary structure, is solid area and surface area ratio in contact surface.r1It is level-one knot
The roughness value of structure, r2It is the i.e. practical gross area of roughness value and surface area ratio of secondary structure, ɑ1And ɑ2It is one respectively
Each column array section length of side in grade, secondary structure, rbRadius of circle is contacted for drop and two level micro-nano structure surface；x1For nanometer
Invasive depth under structure, x2For the invasive depth under micrometer structure.
(3) the wetting state I of current heavy-fluid drop is calculatedf, contact radius of circle rb-f, infiltration of the drop in primary structure
Depth x1-fWith penetration depth x of the drop in grade level structure2-f。
31. the optimization of Local Minimum dimensionless free energy under four kinds of contact conditions：
311. variable：Discrete point coordinates ui, vi, i=1,2 ..., N+1, drop is in primary structure x1With secondary structure x2In
Penetration depth.
312. constraint:
i.
ii.
iii.0≤x1< h1,0≤x2< h2 (45)
Iv. because contour curve function u=f (v) is a convex function, so
v.
Wherein, hɑmaxIt is the maximum height of drop, V is the volume of drop, uiIt is the abscissa of discrete point, viIt is discrete point
Ordinate, rbmaxIt is the maximum value of contact line radius, x1It is penetration depth of the drop in primary structure, x2It is drop two
Penetration depth in level structure, h1It is the height of primary structure, h2It is the height of secondary structure.
313. search:
The search variables x under constraints1,x2,h1,h2, best object is obtained simultaneously using Matlab functions " fmincon "
Calculate the part minimum dimensionless free energy E under four kinds of wetting contact statesI', I=1,2,3,4.
32. the optimization of whole minimum dimensionless free energy under four kinds of contact conditions：Using Matlab functions " fmincon " come
The minimum dimensionless free energy E in search partImin', I=1,2,3,4, in whole minimum dimensionless free energy Emin', constantly count
It calculates, until whole minimum dimensionless free energy difference converges to 10-4Tolerance interval in.At this moment dimensionless free energy is considered as
It is whole minimum dimensionless free energy, and writes down the free E of corresponding whole minimum dimensionlessmin' can when discrete point, and pass through it
Can obtain contour curve and surface contact angleAnd corresponding wetting state IfJustify with contact
Radius rb-f, penetration depth x of the drop in primary structure1-fWith penetration depth x of the drop in grade level structure2-f.Otherwise, add
Times isodisperse N, carries out step 311 again, and 312 and 313.
(4) setting initial volume increment is Δ V=0.5V, and contact radius of circle is fixed and u1=rb=rb-f, x1=x1-f, meter
Calculate corresponding minimum dimensionless free energy EV+ΔV-fix', i.e. EImin'.If u1=rb, v1=0, because of I, rb,x1And x2For definite value when,
So minimum dimensionless free energy is optimized for:
(a) variable:Discrete point coordinates ui, vi, i=2 ..., N+1；
(b) setting optimization object EI'minWith constraints i, ii, iii, iv and v, with step 312.
(c) the search variables u under constraintsi,vi, i=2 ... .., N+1, under the conditions of contact circle is fixed, utilize
Fmincon functions in Matlab obtain optimal object and calculate whole minimum dimensionless free energy EImin'.Constantly counted
It calculates, until whole minimum dimensionless free energy difference converges to 10-4Tolerance interval in, dimensionless free energy at this moment is considered as
It is whole minimum dimensionless free energy EImin', i.e., the corresponding minimum dimensionless free energy of heavy-fluid drop that volume is Δ V+V
EV+ΔV-fix'.Write down discrete point p simultaneouslyi-f[u0,v0] and pi-f[ui,vi], obtain corresponding contour curve and corresponding surface
Contact angle passes through formulaOtherwise, isodisperse N is just doubled, repeats step (b), (c).
(5) when contact radius of circle change when, i.e., contact circle variation when, Local Minimum dimensionless free energy from become EV+ΔV-fix’
For EV+ΔV', step (3) is repeated, E is calculatedV+ΔV'=Emin’。
(6) CAH be as contact circle on energy barrier caused by, can be by the way that E be calculatedbarr=U0l0+U1l1+U2l2,
In formula, EbarrIt is energy barrier, U0, U1And U2It is the energy barrier between drop and substrate, primary structure and secondary structure respectively；
l0, l1, l2It is the physical length between drop and substrate, primary structure and secondary structure respectively.When contact circle variation, part is most
Small dimensionless free energy is from EV+ΔV-fix' become EV+ΔV'.Δ E=EV+ΔV-fix’-EV+ΔV'>0, it is found that when contact circle variation, connects
Tactile circle obtains potential energy Δ E.As Δ E=EbarrWhen, contacting circle can forward, this ACA for being is exactly θɑ。
(7) by using binary system iterative method, Δ V and corresponding θ be may search forɑ=θV+ΔV-fixMake EV+ΔV-fix’-
EV+ΔV'=Ebarr。
(8) setting initial volume increment is Δ V=-0.5V and calculates θ using identical methodr, and calculate contact
Angular lag CAH=θɑ-θr。
Embodiment：
The geometric structure diamete parameter of secondary structure is：The nanostructure square column length of side is ɑ1The week of=100nm, regular square formation
Phase distance b1The height h of=88nm, square column1=400nm, the micrometer structure square column length of side is ɑ2=10 μm, period of regular square formation away from
From b2=8.26 μm, the height h of square column2=13 μm.Intrinsic contact angle of the drop in primary structure, secondary structure and substrate be
θe0=θe1=θe2=106 °, the energy barrier between drop and substrate, primary structure and secondary structure is U0=U1=U2=4.2 ×
10-5。
It is illustrated in figure 5 the variation diagram for being CAH with Bond numbers (Bo), i.e., the corresponding CAH values of different volumes.Illustrate we
The feasibility that method calculates the coarse surface contact angle of secondary structure solid and drop profile, simultaneously for different liquid and is drilled
The second order structure of change is all applicable.
The foregoing is merely one of present invention embodiments, are not intended to limit the scope of the present invention.
Any modification, equivalent replacement, improvement and so on all within the spirits and principles of the present invention are all contained in the protection of the present invention
In range.
Claims (1)
1. a kind of computational methods of the double roughness surface heavy-fluid drop contact angle hysteresis of horizontal rule, include the following steps：
(1) assume a model by drop and micro-nano secondary structure surface composition, wherein first order nanostructure and the second level
Micrometer structure is all square column, and geometric scale periodic distribution, secondary structure and substrate are made of different materials.The summary table of model
Sight area is length of side L0Square, the length of side be much larger than liquid-drop diameter.a1, b1, h1And x1It is the side of the first level structure square column respectively
Length, spacing, height and invasive depth, a2, b2, h2And x2It is the length of side, spacing, height and the infiltration of the second level structure square column respectively
The size of depth, the first level structure is much smaller than the second level structure.Since the heavy-fluid on double roughness surfaces of horizontal rule drips shape
Shape is axisymmetric, so considering this problem on two dimensional surface.It is v axis to select the symmetry axis that heavy-fluid is dripped, heavy-fluid drop
Surface contact surface is u axis, and the contour curve that heavy-fluid is dripped is divided into N equal portions along the directions v using finite difference calculus, is obtained discrete
Point：
Wherein, haIt is the maximum height of drop, uiIt is the abscissa of discrete point, viIt is the ordinate of discrete point.
The derivation of the system dimensionless free energy function of (2) four kinds of different wetting contact conditions, derivation are as follows：
21. dropping in the penetration depth in primary structure and secondary structure by heavy-fluid, wetting contact state is divided into four kinds, respectively
For W-W, C-W, W-C and C-C states, partitioning standards are as follows：
(a) primary structure and secondary structure are in Wenzel states, i.e. x1=h1, x2=h2, this contact condition is named as W-W
State.
(b) primary structure is in Cassie states or contact condition will be changed into Wenzel states, two level from Cassie states
Structure is in Wenzel states, i.e. 0≤x1<h1, x2=h2, this contact condition is named as C-W states.
(c) primary structure is in Wenzel states, and secondary structure is in Cassie states or contact condition will be from Cassie shapes
State is changed into Wenzel states, i.e. x1=h1, 0≤x2<h2, this contact condition is named as W-C states.
(d) primary structure and secondary structure are in Cassie states or contact condition and will be changed into from Cassie states
Wenzel states, i.e. 0≤x1<h1, 0≤x2<h2, this contact condition is named as C-C states.
Wherein, x1It is penetration depth of the drop in primary structure, x2It is penetration depth of the drop in secondary structure, h1It is one
The height of level structure, h2It is the height of secondary structure.
22. the system free energy E of four kinds of different wetting contact conditionsIEqual to EI=Ea+Eb, I=1,2,3,4.Subscript I=1,2,3,
4 indicate W-W, C-W, W-C and C-C state respectively.EaAnd EbRespectively represent gravitional force and surface free energy.
EaIt can be expressed as：
EbIt can be expressed as：
Eb=γLVSLV+γSL0SSL0+γSL1SSL1+γSL2SSL2+γSV0SSV0+γSV1SSV1+γSV2SSV2 (3)
Wherein, ρ is fluid density, and g is acceleration of gravity, and S is the contact area at interface, and γ is interfacial tension coefficient, subscript LV,
SL and SV respectively represents liquid-gas, solid-liquid and solid-air interface, and subscript 1,2 and 0 respectively represents primary structure, secondary structure and base
Bottom.
The external surface area S of dropextIt can be expressed as：
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop.
Drop and the contact surface area of substrate can be expressed as：
Sbase=π rb 2 (5)
In formula, rbRadius of circle is contacted for drop and two level micro-nano structure surface.
According to the difference of each wet face state, can be easy to acquire the S under four kinds of statesLV, SSV0, SSV1, SSV2, SSL0, SSL1,
SSL2, corresponding expression formula:
(a) W-W states:
SLV=Sext=2 π R2(1-cosθ) (6)
SSL0=Sbase(1-f1)(1-f2)=π rb 2(1-f1)(1-f2) (7)
SSL1=Sbaser2[r1-(1-f1)]=π rb 2r2(r1-1+f1) (8)
SSL2=Sbase(1-f1)[r2-(1-f2)]=π rb 2(1-f1)(r2-1+f2) (9)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown drop
External surface area, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micrometer structure
Roughness factor, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For the side of rectangular shaped surface area
It is long, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micrometer structure
Under invasive depth.
(b) C-W states
SSL0=0 (14)
SSL2=0 (16)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown drop
External surface area, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micrometer structure
Roughness factor, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For the side of rectangular shaped surface area
It is long, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micrometer structure
Under invasive depth.
(c) W-C states：
SSL0=0 (21)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown drop
External surface area, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micrometer structure
Roughness factor, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For the side of rectangular shaped surface area
It is long, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micrometer structure
Under invasive depth.
(d) C-C states：
SSL0=0 (28)
SSL2=0 (30)
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, SextFor spherical crown drop
External surface area, SbaseFor the contact area of drop and substrate, r1For the roughness factor of nanostructure, r2For micrometer structure
Roughness factor, f1For the area fraction of nanostructure, f2For the area fraction of micrometer structure, L0For the side of rectangular shaped surface area
It is long, rbRadius of circle, x are contacted for drop and two level micro-nano structure surface1For the invasive depth under nanostructure, x2For micrometer structure
Under invasive depth.
It brings parameter into formula and can obtain the system free energys of four kinds of different wetting contact conditions and be：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
In formula, u is the abscissa of discrete point, and v is the ordinate of discrete point, haIt is the maximum height of drop, f1It is primary structure
Solid portion, f2It is the solid portion of secondary structure, is solid area and surface area ratio in contact surface.r1It is primary structure
Roughness value, r2The roughness value of secondary structure, i.e., the practical gross area and surface area ratio；ɑ1And ɑ2Be respectively level-one,
Each column array section length of side, r in secondary structurebRadius of circle is contacted for drop and two level micro-nano structure surface；x1For nanostructure
Under invasive depth, x2For the invasive depth under micrometer structure.
23. the system dimensionless free energy E of four kinds of different wetting contact conditionsI' determination, I=1,2,3,4 respectively represent W-W,
C-W, W-C and C-C state：When given heavy-fluid drop and the physical property of double rough surfaces,It is constant, and by Young's equation
Equivalent contact angle, which can be obtained, isThe system free energy of four kinds of different wetting contact conditions can be reduced to：
(a) W-W states
(b) C-W states
(c) W-C states
(d) C-C states
(3) the wetting state I of current heavy-fluid drop is calculatedf, contact radius of circle rb-f, penetration depth of the drop in primary structure
x1-fWith penetration depth x of the drop in grade level structure2-f。
The optimization of Local Minimum dimensionless free energy under 31 4 kinds of contact conditions：
311. variable：ui, vi, i=1,2 ..., N+1, drop is in primary structure x1With secondary structure x2In penetration depth；
312. constraint:
i.
ii.
iii.0≤x1< h1,0≤x2< h2 (45)
Iv. because contour curve function u=f (v) is a convex function, so
v.
Wherein, hɑmaxIt is the maximum height of drop, V is the volume of drop, uiIt is the abscissa of discrete point, viIt is the vertical of discrete point
Coordinate, rbmaxIt is the maximum value of contact line radius, x1It is penetration depth of the drop in primary structure, x2It is drop in two level knot
Penetration depth in structure, h1It is the height of primary structure, h2It is the height of secondary structure；
313. search:
The search variables x under constraints1,x2,h1,h2, obtain best object using Matlab functions " fmincon " and calculate
Part minimum dimensionless free energy E under four kinds of wetting contact statesImin', I=1,2,3,4；
32. the optimization of whole minimum dimensionless free energy under four kinds of contact conditions：It is searched for using Matlab functions " fmincon "
The minimum dimensionless free energy E in partImin', I=1,2,3,4, in whole minimum dimensionless free energy Emin', it constantly calculates, directly
10 are converged to whole minimum dimensionless free energy difference-4Tolerance interval in；At this moment dimensionless free energy is taken as whole
Body minimum dimensionless free energy, the wetting state I of current heavy-fluid drop under meterf, contact radius of circle rb-f, drop is in primary structure
Penetration depth x1-fWith
Penetration depth x of the drop in grade level structure2-f；Otherwise, isodisperse N is doubled, carries out step 31 and step 32 again.
(4) setting initial volume increment is Δ V=0.5V, and contact radius of circle is fixed and u1=rb=rb-f, x1=x1-f, calculating pair
The minimum dimensionless free energy E answeredV+ΔV-fix', i.e. EImin'.If u1=rb, v1=0, because of I, rb,x1And x2For definite value when, so
Minimum dimensionless free energy is optimized for:
(a) variable:Discrete point coordinates ui, vi, i=2 ..., N+1；
(b) setting optimization object EI'minWith constraints i, ii, iii, iv and v, with step 312.
(c) the search variables u under constraintsi, vi, i=2 ... .., N+1 are utilized under the conditions of contact circle is fixed
Fmincon functions in Matlab obtain optimal object and calculate whole minimum dimensionless free energy EImin'.Constantly counted
It calculates, until whole minimum dimensionless free energy difference converges to 10-4Tolerance interval in, dimensionless free energy at this moment is considered as
It is whole minimum dimensionless free energy EImin', i.e., the corresponding minimum dimensionless free energy of heavy-fluid drop that volume is Δ V+V
EV+ΔV-fix'.Write down discrete point p simultaneouslyi-f[u0,v0] and pi-f[ui,vi] and obtain corresponding contour curve and corresponding surface
Contact angle passes through formulaOtherwise, isodisperse N is just doubled, step (b), (c) are repeated.
(5) when contact radius of circle change when, i.e., contact circle variation when, Local Minimum dimensionless free energy from become EV+ΔV-fix' be
EV+ΔV', step (3) is repeated, E is calculatedV+ΔV'=Emin’。
(6) CAH be as contact circle on energy barrier caused by, can be by the way that E be calculatedbarr=U0l0+U1l1+U2l2, formula
In, EbarrIt is energy barrier, U0, U1And U2It is the energy barrier between drop and substrate, primary structure and secondary structure respectively；l0,
l1, l2It is the physical length between drop and substrate, primary structure and secondary structure respectively.When contact circle variation when, Local Minimum without
Dimension free energy is from EV+ΔV-fix' become EV+ΔV'.Δ E=EV+ΔV-fix’-EV+ΔV'>0 it is found that when contact circle variation, and contact is justified
Obtain potential energy Δ E.As Δ E=EbarrWhen, contacting circle can forward, this ACA for being is exactly θɑ。
(7) by using binary system iterative method, Δ V and corresponding θ be may search forɑ=θV+ΔV-fixMake EV+ΔV-fix’-EV+ΔV’
=Ebarr。
(8) setting initial volume increment is Δ V=-0.5V and calculates θ using identical methodr, and calculate contact angle hysteresis
CAH=θɑ-θr。
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