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Viscous flow in planar capillaries of variable cross-section
Colloids and Surfaces, 17 (1986) Elsevier Science Publishers B.V.,
273-282 Amsterdam
VISCOUS FLOW IN PLANAR CROSS-SECTION
A.A.
VAVKUSHEVSKII,
Institute
of Physical
(Received
20 June
V.V.
1985;
Academy
accepted
Printed
CAPILLARIES
ARSLANOV
Chemistry,
273 -
and V.A. of Sciences
5 September
in The Netherlands
OF VARIABLE
OGAREV of the USSR,
Moscow
(U.S.S.R.)
1985)
ABSTRACT A method has been developed to study the flow of liquids through planar capillaries made of various materials, the spacing between the walls being from several micrometers down to 0.05 pm. The flow of polydimethylsiloxane (PDMS) and epoxy resin ED-20 into spacings between glass and polymethylmethacrylate surfaces was studied at various temperatures and air humidities. An anomalous decrease of PDMS viscosity when flowing between glass walls was detected which is related to a boundary low-viscosity PDMS layer produced due to water adsorbed on the glass.
INTRODUCTION
From previous studies it is known [ 1, 21 that the structure and properties of thin layers of a liquid can differ from those of the bulk material. Since the dimensions of boundary regions with distorted properties are small, the above-mentioned effects have, until now, been rather poorly investigated. Of great interest is the study of the hydrodynamics of near-the-wall surface layers in polymer melts, because the peculiar properties of such liquids can result in some anomalies even in a thick layer. The viscosity of lubrication-oil films has been determined [3] using the blow-off method. For very thin films (-1000 A), the viscosity was found to be 10 times that of the bulk. The same method was used to study polydimethylsiloxane (PDMS) films of various molecular weights on glass and metal substrates, the viscosity being found to decrease with thickness. The near-the-surface liquid layers can be investigated by observing them flowing in thin capillaries whose thickness is commensurable with the layer’s thickness [4]. However, homogeneous thin (- 1 pm) capillaries are rather difficult to produce [5] and very few materials are suitable for this purpose. Moreover, this method cannot be applied to very viscous liquids, such as polymer melts. In the present work, a new approach is proposed that is based on observations of the motion of a liquid front flowing into a variable-thickness quasi-planar spacing between two surfaces with a known geometry. 0166-6622/86/$03.50
o 1986
Elsevier
Science
Publishers
B.V.
274 EXPERIMENTAL
The flow of the liquids was studied using a cell consisting of a planoconvex lens in contact with a glass plate (for schematics see Fig. 1). In order to compensate for the elastic deformation of glass at the contact point, a special clip was used which allowed the force pressing the lens to the plate to be adjusted. Newton rings in the light of a mercury lamp were used to check the position of the lens with respect to the plate, the glass deformation at the contact point was thereby kept constant within an accuracy of 0.01-0.05 pm. The assembled cell was placed inside a chamber which was tightly sealed by a flange, which contained a needle for inserting the liquid, and thermostatted on a microscope table. The desired chamber humidity was attained using special desiccants. The positions of the liquid front and of the Newton rings were measured by an ocular micrometer with an accuracy of f 1 pm. In our experiments, the radius of curvature, R, of the lens and the distance from its edge to the contact point, x0, were determined in order to study the liquid flowing into a space of lo-O.05 pm (R = 8 cm, x0 = 1 mm).
Fig. 1. Schematics
of the cell consisting
of a lens and a plate. For details see text.
In Fig. 2 the distribution of flow-lines in the central part of the cell is depicted. This was obtained using particles of finely dispersed soot suspended in polydimethylsiloxane (PDMS). As can be seen in Fig. 2, the curvature of the flow-lines in the lens-plate spacing increases with increasing distance from the center. However, near the symmetry axis of the system, the flow-lines are slightly condensed, hence the flow-in problem in the central part can be considered to be one-dimensional. This consideration is, evidently, only valid if either the radius of curvature of the lens or the ratio R/x0 is large. In order to derive the equations of motion for the front of a liquid flowing into the lens-plane spacing let us consider a narrow (width a) duct in the liquid (see Fig. 1). Within this duct we take a region with length dx and use the expression for the force of viscous resistance in an infinitely wide plane spacing [6] (here, we consider the spacing as a plane, since the relative slope of the walls is small; in our experiments x,/h > 500, h being the spacing height).
275
Fig. 2. Flow-lines in the central part of the cell: (1) the point of lens-to-plate (2) Newton rings; (3) the front of the liquid; (4) flow-lines; (5) the lens edge.
contact;
dx
(1)
where n is the viscosity, at the point x;
dx is the spacing’s length, h, is the spacing’s height
-dF,
= 6~7 v,,
h, = R -
+
x
(R2 - xz)1/2
(2)
for a spherical lens, and vaV is the average flow velocity defined from the flux through the S = ah, surface. If lateral in- and out-fluxes are absent, v,, at x is inversely proportional to the spacing’s height
vx =
vT
-
hT hx
(3)
where vT and hT are, respectively, the velocity and the height near the liquid front. We find the force of viscous resistance over all the wetted part of the duct by substituting Eqns (2) and (3) into Eqn (1) and integrating from x0 to xr, xT being the coordinate of the front
-
Fq
s
dF7, =
0
XT s XC,
6dw’T
As x is small compared -FT,
= 8Qah,VT
R2
(.R +
gF=?) X4
dx
(4)
with R we obtain 1
7
XT
--&X”o
(5)
276
We adopt the Laplaceian pressure per unit area of the meniscus to be the main factor, this pressure is due to the formation of a meniscus at the liquid front F, = 2 au cos 0
(6)
Here, u is the surface tension of the liquid, and 0 is the contact angle. In what follows we take F, to be constant, i.e. all the parameters on which it depends are independent of the flow velocity and spacing thickness. If this assumption was not valid, the result would be a slowing down of the liquid in the initial stage of flow (for large ur). The flow equation, -F, = F, , with regard to Eqns (5) and (6) is 8 R2vhTvT
(;-+)
=2acos6
Using substitutions, hT = xk/2R and UT = -dxr/dt, sponding transformations
(7) we obtain after corre-
(8) Integration of Eqn (8) between [OJ] and [x0, xt] gives the equation that determines the position of the liquid front xt at moment t
(9) where 7. is the dimensionless parameter which characterizes the front position at t. A dependence of the type of Eqn (9), for a wedge-shaped spacing formed
Fig. 3. The wedge-shaped
cell. For details see text.
277
from two planes (Fig. 31, can easily be obtained of Eqn (2), the expression h,
=ho
in a similar way if, instead
”
(10)
x0
which relates the spacing thickness Eqn (1).
is substituted
into
2
3 rl x’o t=
with the x coordinate,
2h,o cos I!?
(11)
It should be recalled that Eqns (9) and (11) were derived under the assumption that the spacing walls are parallel in the region & and that no lateral influx exists. Therefore, they can only be used for large values of a/h and x0/h, and the experimental parameters must be chosen accordingly. RESULTS
AND
DISCUSSION
The method proposed allows the flowing of a liquid into planar capillaries made of various materials to be studied even in the case when one of the walls is opaque. A very wide range of capillary thicknesses, from several micrometers down to the values limited by the surface roughness and by deformations in the contact point (-500 A), can be studied in a single experiment. This method can also, in principle, be used if either the shape of the capillary is known a priori or it can be determined by some other methods (interferometric, ellipsometric) in the course of the experiments. Even thinner capillaries can be investigated if a liquid flows into a spacing between walls made of split laminated materials, such as mica. Since the surface of split mica is said to be molecularly smooth, the constraints engendered by surface roughness are then removed and one can observe experimentally how a liquid flows into narrow spacings.
+-&_ Fig. 4. The cell of split mica. See text.
278
The spacing height in such a system must correspond to the deflection of a console with one fixed and one free end [7] (Fig. 4). h, =2
(;)2
(3-E)
(12)
Substitution of Eqn (12) into Eqn (1) and consequent integrations and transformations result in a cumbersome and, hence, hardly applicable expression for the time t required for a liquid to reach the point it. However, by neglecting some small terms, a simplified formula can be derived which is valid for the initial stage of the inflow at x~/x~= 1 and for the final stage at x~/x~< 0.2
Using the method described, we have studied the flow of epoxy resin
Fig. 5. The dependence of T” on time t: (1) epoxy PDMS in a polymethylmethacrylate capillary.
resin ED-20
in a glass capillary;
(2)
279
ED-20 (ER) and PDMS into a cell consisting of a planoconvex lens and a plate. Curve 1 of Fig. 5 shows the time dependence of the dimensionless parameter, rO, characterizing the ER front position in a glass capillary, The value of r,, was calculated by substituting the measured length of the wetted capillary into Eqn (9). The similar dependence for PDMS flowing into a polymethylmethacrylate capillary is given in Fig. 5, curve 2. In the two cases the dependences are clearly seen to be linear in the range of capillary thickness from several micrometers down to about 1500-2000 A, the minimum thickness being determined by surface roughness. This linear behavior indicates the Newtonian character of ER and PDMS flow with constant contact angle 0 at any thickness involved. However, when PDMS flows into thin plane glass capillaries, the experimental points deviate from the theoretical curve. In order to reveal the cause of this anomaly, complex studies were carried out in which several parameters were varied, viz. the polymer molecular weight (M.W. 7.5*104, 14.10’ and 22*104), temperature (from 20 to llO”C), and air humidity (from 5. 10m4 to 100%). It was found that the enhancement of the flow velocity over the theoretically expected values takes place at lower temperatures and molecular weights and at higher relative humidities. The observed anomaly of PDMS flow in the presence of humid air is probably due to variations of the meniscus contact angle which increasingly deviates from the equilibrium value the faster the meniscus moves. Such anomalies could result in a slower flowing of the liquid in the beginning of the experiments. However, as can be seen in Fig. 6, the experimental points fit the curve well at the beginning but subsequently deviate from the curve, the value of cos 0 calculated by substituting the PDMS volume viscosity 77 into Eqn (9) being close to unity. Therefore, the considerable acceleration of inflow shown in the final part of the curves of Fig. 6 cannot be attributed to variations of the contact angle. To determine the cause of this anomaly we have made numerical differentiations of the r,,-t curves, thus obtaining
Fig. 6. The r,--t curves for PDMS (M.W. = 75000) (l), 100%; (2), 60%; (3), 8%; (4), 4%; (5), 5.10.‘%.
for different
values of air humidity:
280
the effective PDMS viscosity (n*) at various capillary thicknesses. Each of the resulting curves presented in Figs 7 and 8 contains two regions with different slopes. The transition point between the two regions of a curve lies at 0.4 < h < 0.9 pm, depending on the experimental conditions. Therefore, we can assume that the layer of flowing PDMS has a complicated structure, viz. boundary PDMS layers with decreased viscosity near the capillary walls with “common” PDMS between them. When the liquid front approaches the point of contact between the lens and the plate, h decreases and the boundary layers come nearer to one other. Then the layers’ contribution increases proportionally with the decrease in capillary thickness; in Figs 7 and 8 it corresponds to the region of smooth decrease of n* at large h. When the front reaches the point where the spacing is twice as thick as the PDMS surface layer, the layers overlap and n* decreases sharply due to the interaction of layers. The structure and formation of the PDMS boundary layers can be explained by hydration of the polymer Si-0-Si bond by the water adsorbed on the glass. The hydration results in an uncoiling of the PDMS globulae with the formation of extended spirals. Because the spirals are oriented in the flow, such a layer would possess lower viscosity than that of the bulk. Such phenomena have been detected in previous studies [8, 91 where the PDMS monolayers and films on water surfaces were investigated. The critical thickness of a PDMS layer was found to be about 2000-3000 A [9] ; in
1
2
3
4
5 h,v
Fig. 7. The dependence of the effective viscosity lary thickness for different values of air humidity:
n* of PDMS (M.W. = 75000) (l), 100%; (2), 60%; (3) 8%.
on capil-
Fig. 8. The dependence of the effective viscosity of n * of PDMS for different values of molecular weight at an air humidity of 100%: (l), M.W. = 220000; (2). M.W. = 140000; (3) M.W. = 75000.
281
thinner layers the viscosity decreases. This critical value is close to that observed in our study (2000-4500 A). On the basis of the proposed model the following interpretation of the experimental data on PDMS flowing into capillaries can be given. The increasing deviations from the straight line 70- t with increasing relative humidity are due to the development of PDMS boundary layers caused by the increased amount of water adsorbed on the glass. The deviations increase with decreasing polymer molecular weight because the shorter spirals readily orient in the flow and the macromolecules are less interlaced. CONCLUSION
In the above we have considered the effects connected with the transition range of polymer film thickness where viscosity depends on h. But at h < 1500-2000 A the location of the liquid front becomes indeterminate because the liquid splits into separate streams which accelerate to the lensto-plate contact point. This effect cannot be related to the geometrical and energetic inhomogeneity of the glass surface, since even when the roughness was increased lo-fold no splitting was detected at h - 1 pm, only the flow velocity decreased. Moreover, in experiments with epoxy resin ED-20 in a glass capillary and with PDMS in a lens-plate polymethylmethacrylate cnpil!ary. only weak deceleration of the front was observed at h - 15002000 A at any air humidity. The splitting of the PDMS front in a glass capillary is, seemingly, also connected with anomalous properties of the thin polymer layer and this effect needs to be examined further.
4
!
2.5
2.9
3.3
T-’ x IO3
Fig. 9. The temperature dependence tivation energy of viscous flow: (l), 75000.
K-’
of the PDMS viscosity used to determine M.W. = 220000; (2), M.W. = 140000; (3),
the acM.W. =
282
In conclusion it should be noted that the method proposed herein is reliable both theoretically and experimentally. Its reliability is verified by the measurements of the viscosity (77) of PDMS for the case of large h when Newtonian flow takes place. In this range of h the PDMS viscosity was calculated for various temperatures and for three different molecular weights, the results being in agreement with those obtained by independent methods. Also, the apparent activation energy of PDMS flow was calculated using the dependencies shown in Fig. 9. For all the PDMS examined the activation energy was about 12 kJ mall’, corresponding to the data of Ref. [lo] . REFERENCES 1
2
3 4 5 6 7 8 9 10
B.V. Deryagin, V.V. Karasev, I.A. Lavygin, 1.1. Skorokhodov and E.N. Khromova, in Yu. S. Lipatov (Ed.), Macromolecules at Boundaries, Naukova Dumka, Kiev, 1971, pp. 139-146 (in Russian). B.V. Deryagin, V.V. Karasev, I.A. Lavygin, 1.1. Skorokhodov and E.N. Khromova, in Yu. S. Lipatov (Ed.), Composite polymer materials, Vol. 10, Naukova Dumka, Kiev, 1981, pp. 51-61 (in Russian). B.V. Deryagin and V.V. Karasev, in J.H. Schulman (Ed.), 2nd Inter. Congr. on Surface Activity, Vol. 3, Butterworths, London, 1957, p. 531. N.N. Fedyakin, Kolloidn. Zh., 36 (1962) 776. Z.M. Zorin, V.D. Sobolev and N.V. Churaev, Dok. Akad. Nauk SSSR, 193 (1970) 630. N.S. Achercan (Ed.), Spravochnik mashinostroitelya (Machinery handbook), Vol. 2, Maskgis, Moscow, 1954, p. 468. Ibid, Vol. 3, p. 56. W. Noll, Pure Appt. Chem., 13 (1966) 101. V.V. Arslanov, T.N. Ivanova and V.A. Ogarev, Dok. Akad. Nauk SSSR, 196 (1971) 1105. T. Kataoka and S. Ueda, J. Polym. Sci. Polym. Lett. Ed., 4 (1966) 317.
273-282 Amsterdam
VISCOUS FLOW IN PLANAR CROSS-SECTION
A.A.
VAVKUSHEVSKII,
Institute
of Physical
(Received
20 June
V.V.
1985;
Academy
accepted
Printed
CAPILLARIES
ARSLANOV
Chemistry,
273 -
and V.A. of Sciences
5 September
in The Netherlands
OF VARIABLE
OGAREV of the USSR,
Moscow
(U.S.S.R.)
1985)
ABSTRACT A method has been developed to study the flow of liquids through planar capillaries made of various materials, the spacing between the walls being from several micrometers down to 0.05 pm. The flow of polydimethylsiloxane (PDMS) and epoxy resin ED-20 into spacings between glass and polymethylmethacrylate surfaces was studied at various temperatures and air humidities. An anomalous decrease of PDMS viscosity when flowing between glass walls was detected which is related to a boundary low-viscosity PDMS layer produced due to water adsorbed on the glass.
INTRODUCTION
From previous studies it is known [ 1, 21 that the structure and properties of thin layers of a liquid can differ from those of the bulk material. Since the dimensions of boundary regions with distorted properties are small, the above-mentioned effects have, until now, been rather poorly investigated. Of great interest is the study of the hydrodynamics of near-the-wall surface layers in polymer melts, because the peculiar properties of such liquids can result in some anomalies even in a thick layer. The viscosity of lubrication-oil films has been determined [3] using the blow-off method. For very thin films (-1000 A), the viscosity was found to be 10 times that of the bulk. The same method was used to study polydimethylsiloxane (PDMS) films of various molecular weights on glass and metal substrates, the viscosity being found to decrease with thickness. The near-the-surface liquid layers can be investigated by observing them flowing in thin capillaries whose thickness is commensurable with the layer’s thickness [4]. However, homogeneous thin (- 1 pm) capillaries are rather difficult to produce [5] and very few materials are suitable for this purpose. Moreover, this method cannot be applied to very viscous liquids, such as polymer melts. In the present work, a new approach is proposed that is based on observations of the motion of a liquid front flowing into a variable-thickness quasi-planar spacing between two surfaces with a known geometry. 0166-6622/86/$03.50
o 1986
Elsevier
Science
Publishers
B.V.
274 EXPERIMENTAL
The flow of the liquids was studied using a cell consisting of a planoconvex lens in contact with a glass plate (for schematics see Fig. 1). In order to compensate for the elastic deformation of glass at the contact point, a special clip was used which allowed the force pressing the lens to the plate to be adjusted. Newton rings in the light of a mercury lamp were used to check the position of the lens with respect to the plate, the glass deformation at the contact point was thereby kept constant within an accuracy of 0.01-0.05 pm. The assembled cell was placed inside a chamber which was tightly sealed by a flange, which contained a needle for inserting the liquid, and thermostatted on a microscope table. The desired chamber humidity was attained using special desiccants. The positions of the liquid front and of the Newton rings were measured by an ocular micrometer with an accuracy of f 1 pm. In our experiments, the radius of curvature, R, of the lens and the distance from its edge to the contact point, x0, were determined in order to study the liquid flowing into a space of lo-O.05 pm (R = 8 cm, x0 = 1 mm).
Fig. 1. Schematics
of the cell consisting
of a lens and a plate. For details see text.
In Fig. 2 the distribution of flow-lines in the central part of the cell is depicted. This was obtained using particles of finely dispersed soot suspended in polydimethylsiloxane (PDMS). As can be seen in Fig. 2, the curvature of the flow-lines in the lens-plate spacing increases with increasing distance from the center. However, near the symmetry axis of the system, the flow-lines are slightly condensed, hence the flow-in problem in the central part can be considered to be one-dimensional. This consideration is, evidently, only valid if either the radius of curvature of the lens or the ratio R/x0 is large. In order to derive the equations of motion for the front of a liquid flowing into the lens-plane spacing let us consider a narrow (width a) duct in the liquid (see Fig. 1). Within this duct we take a region with length dx and use the expression for the force of viscous resistance in an infinitely wide plane spacing [6] (here, we consider the spacing as a plane, since the relative slope of the walls is small; in our experiments x,/h > 500, h being the spacing height).
275
Fig. 2. Flow-lines in the central part of the cell: (1) the point of lens-to-plate (2) Newton rings; (3) the front of the liquid; (4) flow-lines; (5) the lens edge.
contact;
dx
(1)
where n is the viscosity, at the point x;
dx is the spacing’s length, h, is the spacing’s height
-dF,
= 6~7 v,,
h, = R -
+
x
(R2 - xz)1/2
(2)
for a spherical lens, and vaV is the average flow velocity defined from the flux through the S = ah, surface. If lateral in- and out-fluxes are absent, v,, at x is inversely proportional to the spacing’s height
vx =
vT
-
hT hx
(3)
where vT and hT are, respectively, the velocity and the height near the liquid front. We find the force of viscous resistance over all the wetted part of the duct by substituting Eqns (2) and (3) into Eqn (1) and integrating from x0 to xr, xT being the coordinate of the front
-
Fq
s
dF7, =
0
XT s XC,
6dw’T
As x is small compared -FT,
= 8Qah,VT
R2
(.R +
gF=?) X4
dx
(4)
with R we obtain 1
7
XT
--&X”o
(5)
276
We adopt the Laplaceian pressure per unit area of the meniscus to be the main factor, this pressure is due to the formation of a meniscus at the liquid front F, = 2 au cos 0
(6)
Here, u is the surface tension of the liquid, and 0 is the contact angle. In what follows we take F, to be constant, i.e. all the parameters on which it depends are independent of the flow velocity and spacing thickness. If this assumption was not valid, the result would be a slowing down of the liquid in the initial stage of flow (for large ur). The flow equation, -F, = F, , with regard to Eqns (5) and (6) is 8 R2vhTvT
(;-+)
=2acos6
Using substitutions, hT = xk/2R and UT = -dxr/dt, sponding transformations
(7) we obtain after corre-
(8) Integration of Eqn (8) between [OJ] and [x0, xt] gives the equation that determines the position of the liquid front xt at moment t
(9) where 7. is the dimensionless parameter which characterizes the front position at t. A dependence of the type of Eqn (9), for a wedge-shaped spacing formed
Fig. 3. The wedge-shaped
cell. For details see text.
277
from two planes (Fig. 31, can easily be obtained of Eqn (2), the expression h,
=ho
in a similar way if, instead
”
(10)
x0
which relates the spacing thickness Eqn (1).
is substituted
into
2
3 rl x’o t=
with the x coordinate,
2h,o cos I!?
(11)
It should be recalled that Eqns (9) and (11) were derived under the assumption that the spacing walls are parallel in the region & and that no lateral influx exists. Therefore, they can only be used for large values of a/h and x0/h, and the experimental parameters must be chosen accordingly. RESULTS
AND
DISCUSSION
The method proposed allows the flowing of a liquid into planar capillaries made of various materials to be studied even in the case when one of the walls is opaque. A very wide range of capillary thicknesses, from several micrometers down to the values limited by the surface roughness and by deformations in the contact point (-500 A), can be studied in a single experiment. This method can also, in principle, be used if either the shape of the capillary is known a priori or it can be determined by some other methods (interferometric, ellipsometric) in the course of the experiments. Even thinner capillaries can be investigated if a liquid flows into a spacing between walls made of split laminated materials, such as mica. Since the surface of split mica is said to be molecularly smooth, the constraints engendered by surface roughness are then removed and one can observe experimentally how a liquid flows into narrow spacings.
+-&_ Fig. 4. The cell of split mica. See text.
278
The spacing height in such a system must correspond to the deflection of a console with one fixed and one free end [7] (Fig. 4). h, =2
(;)2
(3-E)
(12)
Substitution of Eqn (12) into Eqn (1) and consequent integrations and transformations result in a cumbersome and, hence, hardly applicable expression for the time t required for a liquid to reach the point it. However, by neglecting some small terms, a simplified formula can be derived which is valid for the initial stage of the inflow at x~/x~= 1 and for the final stage at x~/x~< 0.2
Using the method described, we have studied the flow of epoxy resin
Fig. 5. The dependence of T” on time t: (1) epoxy PDMS in a polymethylmethacrylate capillary.
resin ED-20
in a glass capillary;
(2)
279
ED-20 (ER) and PDMS into a cell consisting of a planoconvex lens and a plate. Curve 1 of Fig. 5 shows the time dependence of the dimensionless parameter, rO, characterizing the ER front position in a glass capillary, The value of r,, was calculated by substituting the measured length of the wetted capillary into Eqn (9). The similar dependence for PDMS flowing into a polymethylmethacrylate capillary is given in Fig. 5, curve 2. In the two cases the dependences are clearly seen to be linear in the range of capillary thickness from several micrometers down to about 1500-2000 A, the minimum thickness being determined by surface roughness. This linear behavior indicates the Newtonian character of ER and PDMS flow with constant contact angle 0 at any thickness involved. However, when PDMS flows into thin plane glass capillaries, the experimental points deviate from the theoretical curve. In order to reveal the cause of this anomaly, complex studies were carried out in which several parameters were varied, viz. the polymer molecular weight (M.W. 7.5*104, 14.10’ and 22*104), temperature (from 20 to llO”C), and air humidity (from 5. 10m4 to 100%). It was found that the enhancement of the flow velocity over the theoretically expected values takes place at lower temperatures and molecular weights and at higher relative humidities. The observed anomaly of PDMS flow in the presence of humid air is probably due to variations of the meniscus contact angle which increasingly deviates from the equilibrium value the faster the meniscus moves. Such anomalies could result in a slower flowing of the liquid in the beginning of the experiments. However, as can be seen in Fig. 6, the experimental points fit the curve well at the beginning but subsequently deviate from the curve, the value of cos 0 calculated by substituting the PDMS volume viscosity 77 into Eqn (9) being close to unity. Therefore, the considerable acceleration of inflow shown in the final part of the curves of Fig. 6 cannot be attributed to variations of the contact angle. To determine the cause of this anomaly we have made numerical differentiations of the r,,-t curves, thus obtaining
Fig. 6. The r,--t curves for PDMS (M.W. = 75000) (l), 100%; (2), 60%; (3), 8%; (4), 4%; (5), 5.10.‘%.
for different
values of air humidity:
280
the effective PDMS viscosity (n*) at various capillary thicknesses. Each of the resulting curves presented in Figs 7 and 8 contains two regions with different slopes. The transition point between the two regions of a curve lies at 0.4 < h < 0.9 pm, depending on the experimental conditions. Therefore, we can assume that the layer of flowing PDMS has a complicated structure, viz. boundary PDMS layers with decreased viscosity near the capillary walls with “common” PDMS between them. When the liquid front approaches the point of contact between the lens and the plate, h decreases and the boundary layers come nearer to one other. Then the layers’ contribution increases proportionally with the decrease in capillary thickness; in Figs 7 and 8 it corresponds to the region of smooth decrease of n* at large h. When the front reaches the point where the spacing is twice as thick as the PDMS surface layer, the layers overlap and n* decreases sharply due to the interaction of layers. The structure and formation of the PDMS boundary layers can be explained by hydration of the polymer Si-0-Si bond by the water adsorbed on the glass. The hydration results in an uncoiling of the PDMS globulae with the formation of extended spirals. Because the spirals are oriented in the flow, such a layer would possess lower viscosity than that of the bulk. Such phenomena have been detected in previous studies [8, 91 where the PDMS monolayers and films on water surfaces were investigated. The critical thickness of a PDMS layer was found to be about 2000-3000 A [9] ; in
1
2
3
4
5 h,v
Fig. 7. The dependence of the effective viscosity lary thickness for different values of air humidity:
n* of PDMS (M.W. = 75000) (l), 100%; (2), 60%; (3) 8%.
on capil-
Fig. 8. The dependence of the effective viscosity of n * of PDMS for different values of molecular weight at an air humidity of 100%: (l), M.W. = 220000; (2). M.W. = 140000; (3) M.W. = 75000.
281
thinner layers the viscosity decreases. This critical value is close to that observed in our study (2000-4500 A). On the basis of the proposed model the following interpretation of the experimental data on PDMS flowing into capillaries can be given. The increasing deviations from the straight line 70- t with increasing relative humidity are due to the development of PDMS boundary layers caused by the increased amount of water adsorbed on the glass. The deviations increase with decreasing polymer molecular weight because the shorter spirals readily orient in the flow and the macromolecules are less interlaced. CONCLUSION
In the above we have considered the effects connected with the transition range of polymer film thickness where viscosity depends on h. But at h < 1500-2000 A the location of the liquid front becomes indeterminate because the liquid splits into separate streams which accelerate to the lensto-plate contact point. This effect cannot be related to the geometrical and energetic inhomogeneity of the glass surface, since even when the roughness was increased lo-fold no splitting was detected at h - 1 pm, only the flow velocity decreased. Moreover, in experiments with epoxy resin ED-20 in a glass capillary and with PDMS in a lens-plate polymethylmethacrylate cnpil!ary. only weak deceleration of the front was observed at h - 15002000 A at any air humidity. The splitting of the PDMS front in a glass capillary is, seemingly, also connected with anomalous properties of the thin polymer layer and this effect needs to be examined further.
4
!
2.5
2.9
3.3
T-’ x IO3
Fig. 9. The temperature dependence tivation energy of viscous flow: (l), 75000.
K-’
of the PDMS viscosity used to determine M.W. = 220000; (2), M.W. = 140000; (3),
the acM.W. =
282
In conclusion it should be noted that the method proposed herein is reliable both theoretically and experimentally. Its reliability is verified by the measurements of the viscosity (77) of PDMS for the case of large h when Newtonian flow takes place. In this range of h the PDMS viscosity was calculated for various temperatures and for three different molecular weights, the results being in agreement with those obtained by independent methods. Also, the apparent activation energy of PDMS flow was calculated using the dependencies shown in Fig. 9. For all the PDMS examined the activation energy was about 12 kJ mall’, corresponding to the data of Ref. [lo] . REFERENCES 1
2
3 4 5 6 7 8 9 10
B.V. Deryagin, V.V. Karasev, I.A. Lavygin, 1.1. Skorokhodov and E.N. Khromova, in Yu. S. Lipatov (Ed.), Macromolecules at Boundaries, Naukova Dumka, Kiev, 1971, pp. 139-146 (in Russian). B.V. Deryagin, V.V. Karasev, I.A. Lavygin, 1.1. Skorokhodov and E.N. Khromova, in Yu. S. Lipatov (Ed.), Composite polymer materials, Vol. 10, Naukova Dumka, Kiev, 1981, pp. 51-61 (in Russian). B.V. Deryagin and V.V. Karasev, in J.H. Schulman (Ed.), 2nd Inter. Congr. on Surface Activity, Vol. 3, Butterworths, London, 1957, p. 531. N.N. Fedyakin, Kolloidn. Zh., 36 (1962) 776. Z.M. Zorin, V.D. Sobolev and N.V. Churaev, Dok. Akad. Nauk SSSR, 193 (1970) 630. N.S. Achercan (Ed.), Spravochnik mashinostroitelya (Machinery handbook), Vol. 2, Maskgis, Moscow, 1954, p. 468. Ibid, Vol. 3, p. 56. W. Noll, Pure Appt. Chem., 13 (1966) 101. V.V. Arslanov, T.N. Ivanova and V.A. Ogarev, Dok. Akad. Nauk SSSR, 196 (1971) 1105. T. Kataoka and S. Ueda, J. Polym. Sci. Polym. Lett. Ed., 4 (1966) 317.