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Surface deformation of thin coatings caused by evaporative convection
Surface Deformation of Thin Coatings Caused by Evaporative Convection II. Thermocapillary Flow J. N. ANAND Plastics Fundamental Research Laboratory, The Dow Chemical Company, Midland, Michigan ~8640 T h e autohesive properties of t h i n - c o a t e d surfaces depend on t h e i r texture which in ease of hot solution castings is a consequence of the e v a p o r a t i v e convention flows set up due to solvent removal and e n v i r o n m e n t a l cooling. An expression for the surface d e f o r m a t i o n or t h e final cell depressions caused b y surface t e n s i o n - d r i v e n flow is o b t a i n e d b y considering l a m i n a r t h e r m o c a p i l l a r y flow and a balance between kinetic and p o t e n t i a l energies.
INTRODUCTION The previous paper dealt with a method for macroscopic surface replication developed for mathematical description of surface and experimental study of the mechanism of autohesion. Specifically, the ease of Saran-coatings, east from hot solutions, was discussed. The coatings develop a rough surface having a polygonal cellular texture. The autohesive properties of such surfaces would, thus, depend on their roughness viz. the cell size, and shape since they have to deform to make contact for interracial bonding to take place (1). If the surface roughness were controllable and their shape predictable from casting conditions, it would be possible to impart any desired autohesive characteristics to such coatings. An understanding of the mechanism causing cellular texture is, therefore essential. The final texture, in fact, is the end result of the evaporative convection flow that sets in after casting. Since the casting thicknesses are, usually, very small, the flow is predominantly a surface tension driven type rather than a buoyancy driven (2). Rayleigh's hydrodynamic (3,4), and Pearson's surface tension solutions (5) pre-
diet instability criteria, such as, Rayleigh and Thompson numbers, the critical transitional thickness between the two types of flows, and the cell size in terms of casting conditions and other physical properties of the system. Hershey (7) has obtained an expression for the surface depression of cells at the center according to which the surface depression is inversely proportional to the thickness of film. This expression is applicable for limited thicknesses only and is basically wrong since it gives unlimited deformation when the film thickness is very small. A new expression for surface depression is derived based on the concept of laminar thermoeapillary flow in individual cells and conservation of energy. Thus, on solidification of the film, the kinetic energy of the flowing mass is converted into potential energy resulting in raising of the mass at the cell boundary and a simultaneous lowering at the cell center. T H E R M O C A P I L L A R Y FLOW Surface elevation difference may be derived from the thermocapillary motion theory given by Levich (6). Following as-
Journat of Crinoidand Interface Science, Vol. 31, No. 2, October 1969 203
204
ANAND
sumptions are made: (i) Polygonal cells may be divided into wedge shaped segments in which the flow actually takes place. (ii) The top surface of these wedges may be considered as straight. (iii) The wedges may further be considered to have uniform depth. As shown in Fig. 1, consider one such rectangular cell supporting a lateral temperature difference between the cell center and its wall. This leads to a variation in surface tension from one point to another and consequently results in tangential stress pt in the liquid surface given as pt = grad ~
In the absence of the usual convection flow, the hydrodynamic equations are simplified considerably and we may assume that v v = O.
(4)
The Navier-Stokes equations of motion, therefore, become Op _ ( 02vx 02v~'~ Ox ~ \Ox 2 + O f ] " for shallow depths Ov_~<< Or__!. Ox Oy
(6)
Thus, (5) reduces to 2
(1)
Op _
p~ tends to move the surface from lower to higher surface tension. This motion is known as the capillary convection. The usual bulk thermal convective motion will be very small here since the surface of the liquid is sufficiently large compared to its volume where the surface effects would be predominant. Let the wall temperature be T1 and T2 as shown in Fig. 1 such that
(5)
(7)
o v~
Ox
~ Of "
We may also assume that Op _ O. Oy
(8)
The boundary conditions are (v~)y=~ = O,
and
T2 > T~ Take x and y axes as shown where x is along the surface in the direction of decreasing temperature and y is normal to the surface in the direction of decreasing depth. The variation in surface tension is given by 0¢ a~ = ~-~ AT
(2)
" \ O y ] v=o
The former implies that liquid in contact with the lower solid surface is stationary whereas the later implies that at the surface viscous stress is equal to the surface force. Integrating (7) and using the b.c.'s (9), we get v~ = a +
Substituting (2) in (1), we get
10p +~__~y2,
by
(10)
where pt - Ox
OT a=-
x
TI
10(r
-~
T2 b
//////////f/////////
--
2~ Ox
2 h,
(11)
1 0a ___ I~ Ox "
Thus,
,y FIG. 1. Diagram showing convective flow in a cell. Journal of Colloid and Interface Science,
~
10p
h
V o l . 31, N o . 2, O c t o b e r
1969
10c~
v~ -- -
(h -- y) +
10p
@2 - - h2). ( 1 2 )
SURFACE DEFORMATION CAUSED BY EVAPORATIVE CONVECTION. II
205
This should satisfy the continuity equation 1
h
fov~ dy = O.
l
)-___
(13)
~-)---
/
//
V I_
Thus, we have
foh v~ dy = -l&rh2 ) -]- I~0p ( ~ --~ff
V
t
_I
I-
-I
1 0o- h2 ~Ox2 _
1 0 P h 3_t~:xr~ O~h2 3t~ Ox tt
Ox
(14)
'> ,IV
2t~ Ox 30x
O.
Therefore
Op Ox
3 0o 2h Ox"
-
(15)
Elevation Difference
FIG. 2. Schematic formation of a trapezoidal cross-section from a rectangular cross section.
This is calculated from an energy balance. When the resin solidifies in our ease all the kinetic energy is converted into potential energy which raises the elevation at the side of lower temperatures toward whieh theflow takes place.
The center of gravity of the fluid is raised from h/2 to yo where yc is given by (h - A h ) t ( h -- Ah') + 2 / \
Kinetic Energy
(17)
• (h-Ah+2~)=
A cell of length l, height h and unit width is shown in Fig. 2. Consider a differential element of volume dx dy. If density p is constant, the kinetic energy T of the fluid mass in the cell is h
(2Ah)l
(hl)yo.
Therefore, h yc= ~+
l
(Ah) 2 67 '
(18)
and the potential energy V becomes v = (phl)g Ah2 -6~"
Ox/ ( h - - g ) . . ( h _
y)2 (16) Equating the two energies T and V, we get
1 p(&r/Ox)2h21 180
(19)
phlg(Ah) 2
~2
6h
1 p(O~/Ox)h ~ =
-
-
180
~2
(20)
Potential Energy To calculate the potential energy assume a straight line profile as an approximation as shown in Fig. 2.
Therefore,
(21) ,
\~I
"
Journal of Colloid and Interface Science, Vol. 31, No. 2, October 1969
206
ANAND
Thus, the total difference in elevation is .
(22)
RESULTS AND DISCUSSION
F r o m his experimental observation Benard (2) attributed the surface deformation during convection to the forces of surface tension. Hershey (7) obtained the following expression for the surface depression Ah due to differences in surface tension caused by surface temperature and concentration variations. Ah - 3Aa _ S ( d z / d T ) a T ~ t pgh pgh '
(23)
where Aa is the change in surface tension due to a lateral t e m p e r a t u r e difference of ATlat On a close examination a drawback of this expression becomes obvious. Since Ah cc 1, h we have a limiting value of h below which this expression will not hold. T h i s is obtained when h = Ah
limitation on the range of application of this expression. Variation of Ah as inversely proportional to viscosity is reasonable to expect since a more viscous material will offer higher resistance to deformation and vice versa. There is no such t e r m appearing in Hershey's expression. An experimental verification of this pression is given in the next paper (8). I t m a y be observed t h a t only kinetic and potential energies are considered. There are other energies, such as heat, surface, etc., which one should account for, to be rigorous. Since we are dealing with small changes in surfaces, the effect of the surface energy would be small. However, the thermal effects m a y be large or small, depending upon the casting conditions. We have ignored these effects for the present. L I S T OF SYMBOLS
pt = = T, T1, T2 = x, y =
tangential stress in d y n e / e m 2 surface tension in d y n e / e m temperatures in °C distance in x and y directions in em vx, v:j = velocity components in x and y directions in era/see p = pressure in dyne/era 2 h = height in em = viscosity in dyne sec/cm 2 a, b = constants of integration T = Kinetic energy in dyne em y~ = center of mass coordinate in em p = density in g m / e m 3 g = acceleration due to gravity in em/'sec2 V = potential energy in dyne em ATilt = lateral temperature difference in °C h~i,n = limiting film thickness in cm
Thus,
As will be shown in the next paper (8) this yields a value of h well above the range of thicknesses we deal with in ease of thin coatings. T h e expression derived in this paper based on thermocapillary flow due to evaporative convection seems reasonable. Hence 0~ Ah ~ = - , h , Or,
and
1 -
Thus, nh is greater if surface tension is greater and vice versa. Also Ah is directly proportional to the coating thickness. This is also reasonable since one would expect deformation to be greater for thiek layers and smaller for thin layers. There is, also, no
REFERENCES 1. ANAND, J. N., K•RAM, I t . J., J . A d h e s i o n , 1, 16 (1969).
Journal of Colloid and Interface Science, ¥ol. 31, No. 2, October 1969
SURFACE
DEFORMATION
CAUSED
2. BERG, J. C., "Advances in Chemical Engineer-
ing," (T. B. Drew, J. W. Hoopes, Jr., Eds.) Vol. 6, Academic Press, New York (1966). 3. CHANDRABEKHAR, S., "Hydrodynamic and Hydromagnetic Stability," Oxford Univ. Press, London (1962). 4. LIN, C. C., "The Theory of Hydrodynamic Stability", Cambridge Univ. Press, London (1955).
BY
EVAPORATIVE
CONVECTION.
II
207
5. PEARSON, J. R. A., J. Fluid Mechanics 4, 489 (1958). 6. LEVICH, V. G., "Physiochemical Hydrodynamics," Prentice Hall, Englewood Cliffs (1962). 7. HERSHEY, A. V., Phys. Rev. 56, 204 (1939). 8. ANAND, ft. N. AND KARAM, i .
J., dr. C o l l o i d
Interface Sci. 30, 43rd National Colloid
Symposium (Preprints) 127-134 (1969).
Journal of Colloid and lnterfacv Science, Vo1.31, No. 2, October 1969
INTRODUCTION The previous paper dealt with a method for macroscopic surface replication developed for mathematical description of surface and experimental study of the mechanism of autohesion. Specifically, the ease of Saran-coatings, east from hot solutions, was discussed. The coatings develop a rough surface having a polygonal cellular texture. The autohesive properties of such surfaces would, thus, depend on their roughness viz. the cell size, and shape since they have to deform to make contact for interracial bonding to take place (1). If the surface roughness were controllable and their shape predictable from casting conditions, it would be possible to impart any desired autohesive characteristics to such coatings. An understanding of the mechanism causing cellular texture is, therefore essential. The final texture, in fact, is the end result of the evaporative convection flow that sets in after casting. Since the casting thicknesses are, usually, very small, the flow is predominantly a surface tension driven type rather than a buoyancy driven (2). Rayleigh's hydrodynamic (3,4), and Pearson's surface tension solutions (5) pre-
diet instability criteria, such as, Rayleigh and Thompson numbers, the critical transitional thickness between the two types of flows, and the cell size in terms of casting conditions and other physical properties of the system. Hershey (7) has obtained an expression for the surface depression of cells at the center according to which the surface depression is inversely proportional to the thickness of film. This expression is applicable for limited thicknesses only and is basically wrong since it gives unlimited deformation when the film thickness is very small. A new expression for surface depression is derived based on the concept of laminar thermoeapillary flow in individual cells and conservation of energy. Thus, on solidification of the film, the kinetic energy of the flowing mass is converted into potential energy resulting in raising of the mass at the cell boundary and a simultaneous lowering at the cell center. T H E R M O C A P I L L A R Y FLOW Surface elevation difference may be derived from the thermocapillary motion theory given by Levich (6). Following as-
Journat of Crinoidand Interface Science, Vol. 31, No. 2, October 1969 203
204
ANAND
sumptions are made: (i) Polygonal cells may be divided into wedge shaped segments in which the flow actually takes place. (ii) The top surface of these wedges may be considered as straight. (iii) The wedges may further be considered to have uniform depth. As shown in Fig. 1, consider one such rectangular cell supporting a lateral temperature difference between the cell center and its wall. This leads to a variation in surface tension from one point to another and consequently results in tangential stress pt in the liquid surface given as pt = grad ~
In the absence of the usual convection flow, the hydrodynamic equations are simplified considerably and we may assume that v v = O.
(4)
The Navier-Stokes equations of motion, therefore, become Op _ ( 02vx 02v~'~ Ox ~ \Ox 2 + O f ] " for shallow depths Ov_~<< Or__!. Ox Oy
(6)
Thus, (5) reduces to 2
(1)
Op _
p~ tends to move the surface from lower to higher surface tension. This motion is known as the capillary convection. The usual bulk thermal convective motion will be very small here since the surface of the liquid is sufficiently large compared to its volume where the surface effects would be predominant. Let the wall temperature be T1 and T2 as shown in Fig. 1 such that
(5)
(7)
o v~
Ox
~ Of "
We may also assume that Op _ O. Oy
(8)
The boundary conditions are (v~)y=~ = O,
and
T2 > T~ Take x and y axes as shown where x is along the surface in the direction of decreasing temperature and y is normal to the surface in the direction of decreasing depth. The variation in surface tension is given by 0¢ a~ = ~-~ AT
(2)
" \ O y ] v=o
The former implies that liquid in contact with the lower solid surface is stationary whereas the later implies that at the surface viscous stress is equal to the surface force. Integrating (7) and using the b.c.'s (9), we get v~ = a +
Substituting (2) in (1), we get
10p +~__~y2,
by
(10)
where pt - Ox
OT a=-
x
TI
10(r
-~
T2 b
//////////f/////////
--
2~ Ox
2 h,
(11)
1 0a ___ I~ Ox "
Thus,
,y FIG. 1. Diagram showing convective flow in a cell. Journal of Colloid and Interface Science,
~
10p
h
V o l . 31, N o . 2, O c t o b e r
1969
10c~
v~ -- -
(h -- y) +
10p
@2 - - h2). ( 1 2 )
SURFACE DEFORMATION CAUSED BY EVAPORATIVE CONVECTION. II
205
This should satisfy the continuity equation 1
h
fov~ dy = O.
l
)-___
(13)
~-)---
/
//
V I_
Thus, we have
foh v~ dy = -l&rh2 ) -]- I~0p ( ~ --~ff
V
t
_I
I-
-I
1 0o- h2 ~Ox2 _
1 0 P h 3_t~:xr~ O~h2 3t~ Ox tt
Ox
(14)
'> ,IV
2t~ Ox 30x
O.
Therefore
Op Ox
3 0o 2h Ox"
-
(15)
Elevation Difference
FIG. 2. Schematic formation of a trapezoidal cross-section from a rectangular cross section.
This is calculated from an energy balance. When the resin solidifies in our ease all the kinetic energy is converted into potential energy which raises the elevation at the side of lower temperatures toward whieh theflow takes place.
The center of gravity of the fluid is raised from h/2 to yo where yc is given by (h - A h ) t ( h -- Ah') + 2 / \
Kinetic Energy
(17)
• (h-Ah+2~)=
A cell of length l, height h and unit width is shown in Fig. 2. Consider a differential element of volume dx dy. If density p is constant, the kinetic energy T of the fluid mass in the cell is h
(2Ah)l
(hl)yo.
Therefore, h yc= ~+
l
(Ah) 2 67 '
(18)
and the potential energy V becomes v = (phl)g Ah2 -6~"
Ox/ ( h - - g ) . . ( h _
y)2 (16) Equating the two energies T and V, we get
1 p(&r/Ox)2h21 180
(19)
phlg(Ah) 2
~2
6h
1 p(O~/Ox)h ~ =
-
-
180
~2
(20)
Potential Energy To calculate the potential energy assume a straight line profile as an approximation as shown in Fig. 2.
Therefore,
(21) ,
\~I
"
Journal of Colloid and Interface Science, Vol. 31, No. 2, October 1969
206
ANAND
Thus, the total difference in elevation is .
(22)
RESULTS AND DISCUSSION
F r o m his experimental observation Benard (2) attributed the surface deformation during convection to the forces of surface tension. Hershey (7) obtained the following expression for the surface depression Ah due to differences in surface tension caused by surface temperature and concentration variations. Ah - 3Aa _ S ( d z / d T ) a T ~ t pgh pgh '
(23)
where Aa is the change in surface tension due to a lateral t e m p e r a t u r e difference of ATlat On a close examination a drawback of this expression becomes obvious. Since Ah cc 1, h we have a limiting value of h below which this expression will not hold. T h i s is obtained when h = Ah
limitation on the range of application of this expression. Variation of Ah as inversely proportional to viscosity is reasonable to expect since a more viscous material will offer higher resistance to deformation and vice versa. There is no such t e r m appearing in Hershey's expression. An experimental verification of this pression is given in the next paper (8). I t m a y be observed t h a t only kinetic and potential energies are considered. There are other energies, such as heat, surface, etc., which one should account for, to be rigorous. Since we are dealing with small changes in surfaces, the effect of the surface energy would be small. However, the thermal effects m a y be large or small, depending upon the casting conditions. We have ignored these effects for the present. L I S T OF SYMBOLS
pt = = T, T1, T2 = x, y =
tangential stress in d y n e / e m 2 surface tension in d y n e / e m temperatures in °C distance in x and y directions in em vx, v:j = velocity components in x and y directions in era/see p = pressure in dyne/era 2 h = height in em = viscosity in dyne sec/cm 2 a, b = constants of integration T = Kinetic energy in dyne em y~ = center of mass coordinate in em p = density in g m / e m 3 g = acceleration due to gravity in em/'sec2 V = potential energy in dyne em ATilt = lateral temperature difference in °C h~i,n = limiting film thickness in cm
Thus,
As will be shown in the next paper (8) this yields a value of h well above the range of thicknesses we deal with in ease of thin coatings. T h e expression derived in this paper based on thermocapillary flow due to evaporative convection seems reasonable. Hence 0~ Ah ~ = - , h , Or,
and
1 -
Thus, nh is greater if surface tension is greater and vice versa. Also Ah is directly proportional to the coating thickness. This is also reasonable since one would expect deformation to be greater for thiek layers and smaller for thin layers. There is, also, no
REFERENCES 1. ANAND, J. N., K•RAM, I t . J., J . A d h e s i o n , 1, 16 (1969).
Journal of Colloid and Interface Science, ¥ol. 31, No. 2, October 1969
SURFACE
DEFORMATION
CAUSED
2. BERG, J. C., "Advances in Chemical Engineer-
ing," (T. B. Drew, J. W. Hoopes, Jr., Eds.) Vol. 6, Academic Press, New York (1966). 3. CHANDRABEKHAR, S., "Hydrodynamic and Hydromagnetic Stability," Oxford Univ. Press, London (1962). 4. LIN, C. C., "The Theory of Hydrodynamic Stability", Cambridge Univ. Press, London (1955).
BY
EVAPORATIVE
CONVECTION.
II
207
5. PEARSON, J. R. A., J. Fluid Mechanics 4, 489 (1958). 6. LEVICH, V. G., "Physiochemical Hydrodynamics," Prentice Hall, Englewood Cliffs (1962). 7. HERSHEY, A. V., Phys. Rev. 56, 204 (1939). 8. ANAND, ft. N. AND KARAM, i .
J., dr. C o l l o i d
Interface Sci. 30, 43rd National Colloid
Symposium (Preprints) 127-134 (1969).
Journal of Colloid and lnterfacv Science, Vo1.31, No. 2, October 1969