The Dynamics of Chemically Reacting Liquid Drops on Porous Surfaces William J. Frederick, Jr.,* and Edward G. Bobalek Chemical Engineering Department, University of Maine, Orono, Maine 04401
Initial wetting and spreading of fluids representing adhesives, paints, or coating fluids on microrough surfaces were investigated and modelled. Fluids undergoing physical or chemical changes during spreading were of particular interest. Porous surfaces were constructed by electrodepositing Nylon-talc mixtures onto copper plates. The surface porosity of the coatings obtained varied with coating composition, as determined both by scanning electron microscopy and by spreading rates of nonvolatile liquids. The modelled spreading fluid was a reacting diisocyanate-diamine mixture dissolved in dimethyl phthalate. Reaction kinetics for the system, investigated in bulk and thin films, were found to be independent of reactor geometry. Spreading rates of the reaction mixtures on the Nylon-talc capillary surfaces were measured using moderate speed cinematography. Spreading behavior was explained in terms of a model derived for spreading of pure liquids but modified to account for the time-varying properties of the reaction mixture and for capillary clogging by suspended polymer.
Introduction The spreading rates of pure liquids through thin, porous capillary systems have been investigated and defined. Of practical importance is the spreading behavior over rough surfaces of fluids which more closely resemble adhesives, paints, or coating formulations. The nature of such fluids is that their physical properties change during spreading by solvent evaporation, temperature change, or chemical reaction. The purpose of this research was to determine if the spreading relationships for pure liquids over porous surfaces apply when the fluid properties are not constant, but vary in some predetermined manner with time. Models for P u r e Liquid Spreading The variables controlling the spreading of pure liquids isothermally through porous matrices have been reported by Gillespie (1958), Greinacher (1959), and Cheever (1968). For small drops spreading radially through open slit capillaries, Cheever derived the spreading relationship
where n theoretically has a value of 2. Equation 1 describes steady-state spreading, in which the spreading rate is determined by the balance between capillary and viscous forces. All capillaries inside the spreading front are filled with the spreading liquid. The steady-state spreading region continues until all the liquid present in the drop has entered the capillary system. That is, a central supply of spreading liquid must be maintained. When no central supply of liquid is maintained, liquid spreads by redistributing itself at the leading edge of the spreading drop. Liquid flows from larger capillaries within the boundary of the spreading drop to smaller capillaries outside. The driving force for this redistribution is the capillary pressure difference between capillaries of different sizes, as described by
'
Address correspondence to this author at Polymer and Paper Chemistry Section, BATTELLE. Columbus Laboratories, Columbus, Ohio 43201.
40
Ind. Eng. Chern., Fundarn., Vol. 14, No. 1, 1975
Gillespie has described this case of depletion spreading by the expression
The spreading rate described by eq 3 bears a functional similarity to Cheever's equation, but in addition, it includes a direct dependence on the volume of liquid present. Experimental Procedures Microscopically rough surfaces were constructed by electrodepositing a thin film of microcrystalline Nylonplaty talc from aqueous suspension onto copper plates (Frederick, 1973). These surfaces were employed as the spreading surfaces in all experiments. The porosity level of the surfaces could be varied by varying the ratio of Nylon to talc in the electrodeposited films. Drop spreading rates were measured by two methods. In preliminary experiments with pure liquids, the spreading drop diameter was observed through a 20-power microscope equipped with movable hairline and vernier adjustment and was recorded at fixed time intervals. For later experiments with spreading oils, and for all complex fluid data, the spreading drop diameters were recorded cinematographically. Diameters read from the cine films were converted to actual diameters from a centimeter scale which was recorded simultaneously on the film. Spreading Surface Porosity The porosity level of the electrodeposited Nylon-talc surfaces was characterized both by scanning electron microscopy and by the rate of lateral spreading, under capillary forces, of small liquid drops through the porous surface. In the latter case, eq 1 was used to determine the surface capillary dimension ( d ) from the drop spreading data. The reproducibility of the spreading surface porosity as determined by drop spreading measurements is shown in Table I. On the average, the capillary width is reproducible within d=12%. Also in Table I, the capillary width as measured by electron microscopy is compared with that obtained from drop spreading measurements. Results of the two techniques are in reasonable agreement in all cases.
Table I. Capillary Width Dimensions ( d ) us. '70 Filler in Aviamide-Talc Surfaces as Determined by Several Methods ~
Measurement method
Spreading liquid
Drop spreading, pure liquids %talc
1/ n
d, P
Electron photomicroscopy d,
Drop spreading, homogeneous reacting fluid
P
Number av
Wt av
1l a ~~~
DMP
PDMS, 50 cSt
PDMS, 10 cSt PDMS. 5 cSt
1.0 20.0 40.0 0.79 3.23 3.85 11.7 17.7 20.0 25.0 30.0 33.3 40.0 33.3 33.3
0.463*0.017 0.396i0.007 0.38Di0.005 0.433 f 0 . 0 2 1 0.443 i 0.019 0.431 i 0.024 0.424 i 0.020 0.4083 i 0.015 0.4113 i 0.015 0.391 i 0.014 0.372 i 0.002 0.373 i 0.008 0.37!3 i 0.006 0 . 3 8 1 0.006 0.3863 i 0.009
*
0.242i0.041 0.390+0.041 0.605*0.042 0.129 0 . 0 1 1 0.132 i 0.012 0.113 -ir 0.020 0.129 0.018 0.360 + 0.028 0.347 + 0.056 0.429 0.042 0.623 i 0.045 0.592 f 0.069 0.599 i 0.052 0.493 i 0.012 0.539 i 0.039
0.25i0.08 0.45i0.14 0.55i0.24
0.25+0.01 0.49i0.15 0.65i-0.26
~
d,
P
~~~~~~~~
0.446k0.023 0.393*0.021 0.381*0.026
0.316i0.010 0.405*0.071 0.670*0.128
* *
Limitations of the Drop Spreadling Model As shown in Figure 1, by carefully selecting drop volume, fluid viscosity, and surface porosity, three spreading regions can be observed. The first region (flooding) is simply a transient in which the drop redistributes itself under the influence of gravity. The second, or steady-state region, is that described by Cheever. As seen in Figure 2, a plot of log ( R - Ro) us log ( t )yields a linear relationship, #here the slope equals l / n in eq 1. In all cases observed, n was found to be greater than the theoretical value, 2.0. A plot of l / n us coating composition, Figure 3, clearly indicates that n depends upon the surface composition. An explanation for the difference between the theoretical and experimental values of n will be given presently. In those experiments where the central supply of spreading fluid was depleted, a third (depletion) spreading region was observed. At the onset of the depletion region, all of the spreading fluid is located within the thin, spreading film. From this point, spreading continues as a redistribution of liquid according to capillary size, as predicted by eq 3. When the data are plotted in the form of Gillespie's spreading radius function (R2[R4 - RDo4]) us. time as in Figure 4, the relationship is linear. Since the steady state and depletion regions were of comparable duration, the transition from steady state to depletion spreading is evident, and this explains the curvature occurring at the onset of the depletion region (time zero) in Figure 4. The data do not always pass through zero since averages were used for the time of onset of the depletion region and corresponding spreading radius ( R D o )(See Figure 1).This causes slight shifts along the time axis for individual sets of data. As previously mentioned, measured exponents (n)for eq 1 were found in all cases to be higher than the theoretical value, 2.0 (Table I). Previously published data of Cheever (1968), Greinacher (1959), and Mack (1961) show deviations of comparable magnitude in the same direction. As seen from Table I, n appears to depend upon the coating composition but not on the spreading liquid used. Explanation of this deviation from theory lies in the description of the spreading front. Cheever's model assumed a continuous film of constant average thickness, implying a sharply defined leading edge. In fact, liquid at the lead-
20
' "Depletion" Region
2. 1
gR
"Floaling" Region
I /
I
I
I
I
I
I
5
10
20
50
100
2W
I 5w
Time, Seconds
Figure 1. Typical form of drop spreading curve: 1.0 11. of dimethyl phthalate spreading on 40% talc-Aviamide surface.
I 1
2
5
I I 10 20 Time, Seconds
I 50
I 100
I
2W
Figure 2. Spreading radius vs. time for various polydimethylsiloxanes on 33.3% talc-Aviamine surfaces. Legend: 0 , 5 cSt PDMS; A , 10 cSt PDMS; 0 , 5 0 cSt PDMS.
ing edge does not spread as a film of uniform thickness. Instead, liquid advancement in a zone at the spreading front occurs as filling of smaller, then larger, capillaries, according to eq 2. The selective filling process occurs in a spreading zone of finite width. The width may depend upon both average capillary width and distribution of capillary sizes. Behind this spreading front, flow continues through completely filled capillaries to supply the spreading front. Ind. Eng. Chem., Fundam., Vol. 14, No. 1, 1975
41
0
0 0
0
0
.36 0
1
1
I
10
20
30
'
4
Percent Filler
Figure 3. Reciprocal slope for drop spreading model us. per cent talc in Aviamide-talc spreading surface. Legend: A , 5 cSt PDMS; 0 , l O cSt PDMS; 0 , 5 0 cSt PDMS; 0,DMP.
Depletion Time, Seconds
5
The selective spreading process is similar to the spreading described by eq 3. If selective filling predominated at the leading edge, while steady-state flow occurred in the already filled capillaries, spreading would be slower than predicted by the steady-state model, eq 1. The result would be a higher value of n than predicted by steadystate spreading theory. This result was indeed observed in all cases.
A Complex Spreading Fluid In order to observe the spreading behavior of complex fluids, a spreading fluid with time varying properties was devised. When solutions of p-phenylenediamine (p-9DA) and hexamethylene diisocyanate (HMDI) in dimethyl phthalate (DMP) were mixed, the resulting polymerization altered the fluid properties which govern the spreading behavior. The initial reactant concentration determined the degree of change. A prerequisite for selecting a polymerization reaction to modify the spreading fluid properties was that reaction kinetics of the spreading fluid could be monitored during spreading. Since the spreading drop volume was limited to 10 p1 to reduce the influence of the flooding region, reactant levels of the order of 10-7 equiv had to be determined quantitatively. By using a colorimetric microtechnique which was developed for this system (Frederick, 1973; Snell, 1954) quantities of the unreacted diamine as low as 5 x 10-8 equiv could be measured. The need to monitor reaction rates during spreading arose from the question of the effect of reactor geometry on reaction kinetics. A rate accelerating effect in thin film reactors has been reported by Campbell, et al. (1970), Chae (1962), Moore (1962), and Sunderland (1951). Thin film reaction rates were measured using both the drop spreading system and a rotating disk system as reactors. Batch data were also taken. No reaction rate increase was found with either reactor system, relative to the bulk data (Frederick, 1973). This indicates that the fluid properties as measured in bulk do not, because of a reaction rate effect, differ from those in the spreading film. Properties of the Complex Fluid The polyurea formed from the HMDI-p4DA reaction is only partially soluble in DMP. Therefore, the spreading fluid remained homogeneous for only the first part of the spreading experiments. Insoluble polyureas were formed 42
Ind. Eng. Chern., Fundarn., Vol. 14, No. 1, 1975
4
c 0
1
0
b
250
I
I
500
750
I
I
1OW 1250 Depletion Time, Seconds
I
1500
1750
4. (a) Gillespie spreading radius function (Re - R2R04) us. depletion time for depletion region drop spreading of dimethyl
Figure
phthalate on talc-Aviamide surfaces. Legend: 0 ,20% talc; A, 30% talc; 0 , 40% talc. (b) Gillespie spreading radius function (R6 R2Ro4) us. depletion time for depletion region drop spreading of' dimethyl phthalate on 1%talc-Aviamine surface.
-
after a time interval predictable from initial reactant concentration, subsequently giving a stable suspension. Surface tension, viscosity, and contact angle (on Nylontalc surfaces) of both the homogeneous and heterogeneous mixtures were measured as the reaction progressed. Surface tension and contact angle were unchanged by the progressing reaction, remaining the same as the corresponding values determined for dimethyl phthalate. Viscosity remained constant until the onset of precipitation, after which it increased, depending upon reactant concentration, according to Figure 5. The onset of precipitation, indicated clearly by the viscosity increase, could also be observed as the onset of clouding of the reaction mixture. The times at onset of precipitation for three reactant concentrations are shown in Table 11. Consideration of the mechanism of viscosity increase is useful in explaining the drop spreading behavior of the complex fluids. The reacting mixtures of p-9DA-HMDI in DMP, if undisturbed after precipitation, begin to gel. The gel formed is easily broken by application of shear. Gelation does not occur if sufficient shear is applied continuously during the reaction. For example, the Ostwald viscosity data shown in Figure 5 could be obtained because the fluid was constantly and sufficiently sheared. However, when attempting viscosity measurements in a horizontal capillary as described by Wolpert and Wojtkowiak
,
0
I
0
5
10
15
I
20 Time, Minutes
I
I
I
I
25
30
35
40
Figure 5 . Capillary viscosity us. time for reacting HMDI-p-@DA in DMP a t 25.0"C. Legend: A, 0.1 N initial concentration; 0, 0.15 Ninitial concentration; 0 ,0.2 N initial concentration.
Table 11. Comparison of Data and Models for HMDI/~-ODA in DMP Spreading on Various Talc/Nylon Surfaces Time f o r onset of precipitation as observed in bulk, Reactant Surface relative nor- composi- to initiation mality, tion, of reaction equiv/l. c/o talc (spreading)' 0.10
1 0.15
1 0.20
Capillary clogging parameters for best fit Tu,
sec
tP t
sec
1 20 40
60 (30)
... ... ...
210 210 210
1 20 40
120 (90)
300 600 600
50 90 90
1 20 40
240 (210)
12Ck300 600 300
1 1
1
a In all cases, the reaction was begun 30 sec before the spreading was begun.
(1972), flow ceased shortly after the onset of heterogeneity. The gelling behavior observed indicates that the fluid might best be considered in terms of the gelation process associated with three-dimensional polymers. It is known (Bobalek, et al., 1964) that in three-dimensional gelling systems, micro gel particles begin to form long before the system is actually gelled. Before the final gelation stage is reached, the particles act independently. They are separated enough from each other by a polymer solution so as not to interact with each other. Their behavior is that of a particulate suspension, with increasing viscosity of the suspending phase. Eventually, when enough microgel particles have formed, their interaction with each other causes phase inversion. The flocculated microgel becomes the continuous phase. Analogous behavior is observed with the HMDI-p-QDA in DMP system, and can explain the viscosity behavior, including the maximum occurring a t the highest reactant concentration, observed in Figure 5 . When the reaction
10
20
50
100 2w Time, Seconds
500
1m
2wo
Figure 6. Spreading radius us. time for 0.1 N HMDI-p+DA in DMP on 4070 talc-Aviamide surface. Legend: , --, undisturbed flow model; - - - -, capillary clogging model; tp = 210 sec, T~ = 600 sec ,
reaches a critical point, dependent upon initial reactant concentration, precipitation of the polyurea begins. The mixture remains a stable suspension at all times, however. During the reaction, the viscosity is controlled by the concentration and molecular weight of the dissolved polymer. As the reaction progresses, larger polyurea molecules are formed. As reported by Baker and Bailey (1957), supersaturation of the reactant mixture can be expected. Supersaturation serves to increase the suspending phase viscosity both by increasing the concentration of macromolecules, and allowing the molecular size of the dissolved reactive molecules to increase. Presence of a viscosity maximum is determined by the net rate of formation of dissolved polymer. When the rate of formation of dissolved polymer exceeds the rate of precipitation, the viscosity increases. When the two rates are equal, the viscosity maximum is reached. If the rate of formation never exceeds the rate of precipitation, then a maximum is not obtained. Obviously, supersaturation greatly influences the viscosity behavior. In either case, the suspension viscosity eventually approaches that of a system of gel particles suspended in a saturated polymer solution. Of course, the analogy with the process of true gel formation breaks down after the initial sharp viscosity rise, since phase inversion does not occur. Spreading of a Homogeneous Reaction Mixture Use of the drop spreading model can be made to describe the reacting fluid, before precipitation begins. For the experimental system used (p-@DA-HMDI in DMP), the physical properties governing spreading did not change from those of the solvent until precipitation began. Therefore, for the homogeneous spreading region (of duration 210, 90, and 30 sec for 0.1, 0.15, and 0.2 N initial reactant concentrations, respectively), the solvent spreading model should describe the spPeading rate of the homogeneous reacting fluid. Drop spreading rates were measured by the same procedure as used for simple liquids. The data and model are shown in Figures 6-8. These figures are typical of 28 sets of data taken over the range of three reactant concentrations and three surface compositions. The models for the homogeneous regions are the linear portions of the solid curves at times earlier than those specified above. The agreement between data and model is reasonable in all cases. Upward or downward shifts of all data points for a particular data set indicate the variation in capillary width noted in Table I. Spreading constants (1jn,d) for dimethyl phthalate are Ind. Eng. Chem., Fundam., Vol. 14, No. 1 , 1975 43
10
-
10 Solvent E E w
A
0
w
10
20
50
1W
200 Time, Se:onds
500
1MM
Moo
Figure 7. Spreading radius us. time for 0.15 N HMDI-p-4DA in DMP on 1% talc-Aviamide surface. Legend: --, undisturbed flow model; - - - -, capillary clogging model. A, tp = 90 sec; T~ = 600 sec; B, tp = 50 sec; 7p = 300 sec.
compared with those for the 0.1 N reacting fluid during the homogeneous reaction region in Table I. Agreement for both parameters on three surface compositions is within experimental error. Spreading constants for the fluids with higher reactant concentration were not obtained because of the short spreading time spans and relative scatter of the data.
Flow Models for Gel-Precipitation Systems Spreading continues after precipitation begins, ,but often at a rate slower than that predicted by the homogeneous spreading model. Based on considerations of the physical behavior of the spreading fluid as polymerization occurs, two models for spreading through capillaries after the onset of gelation were developed. They are: (1) spreading of the reacting fluid through pores of unchanging dimensions; (2) spreading of the reacting fluid through a system of shrinking pores, where the gel/precipitate gradually clogs the pores, reducing the effective cross section. Both models assume the form
which is the same as eq 1 but with time varying properties Y L V ( ~ ) , d ( t ) , cos O(t), and q ( t ) . In both models, the capillary pressure exerted on the gelled fluid was assumed to be sufficient to prevent total gelation of the fluid, so that spreading could continue. The models differ in the assumed time dependence of the capillary width, d ( t ) . Mathematical details for both models are shown in the Appendix. The first model is the same as if gelation did not occur. That is, precipitation occurs, but flow of the suspension system continues, with attendant changes in viscosity (Figure 5). Physical properties of the fluid were considered to be those measured in bulk during reaction. This model treats the suspension as if it were a homogeneous fluid not undergoing phase separation during spreading. The second model assumes that the gel particles formed precipitate or coagulate while in the capillaries, reducing the effective capillary cross section available for flow. The selection of a relationship describing the capillary diameter reduction is at best empirical, since the effect of spreading surface area to reactant volume ratio, particle concentration, and capillary size on the clogging rate are not established. However, precipitation in the capillary system, and therefore clogging, may be enhanced by the relatively large surface:fluid volume ratio in the distribut44
Ind. Eng. Chem., Fundam.. Vol. 14, No. 1, 1975
10
20
M
1w
2cQ Time, Seconds
5w
loo0
2wo
Figure 8. Spreading radius
us. time for 0.2 N HMDI-p-@DA in DMP on 20% talc-Aviamide surface. Legend: --, undisturbed flow model;- - - -, capillary clogging model. A, tp = 30 sec; T P = 120 sec; B, tp = sec; T~ = 600 sec.
ed system. This is supported by the work of Baker and Bailey (1957), who reported that by conducting an isocyanate-amine reaction in a solution in which the product urea had been suspended, supersaturation in the early stages of the reaction could be reduced or avoided. This seeding effect could be expected of the Nylon-talc surface also, because of the chemical similarity of amides and ureas, as well as from the physical presence of solid surface. Since the rate of polymer formation is most rapid early in the spreading process, i t is reasonable to assume a capillary blocking rate in which the pores plug fastest initially, as precipitation begins. The model used is an exponential decrease in capillary cross sectional area with time, once precipitation had begun, as shown in eq 5 .
In this form, the pore clogging model depends upon the onset of precipitation (tp) and a time constant ( T ~ ) The . precipitation times observed in bulk for three reactant concentrations are shown in Table II. T ~ however, , is not defined by the bulk fluid properties or by the surface. It might be expected to vary with both surface composition and reactant concentration. Also because of the potential seeding effect of the Nylon-talc surface, the viscosity expression used in the capillary clogging model for the 0.2 N reacting fluid was modified. If the presence of surface did eliminate supersaturation, then the viscosity maximum noted in Figure 5 would not occur. That is, instead of a supersaturated solution-suspension, the solution concentration of polymer would not exceed the saturation concentration. With that assumption, the viscosity of the 0.2 N fluid was assumed to behave as did the 0.1 N and 0.15 N fluids, increasing exponentially to a constant value determined by the saturation concentration of dissolved polymer and the precipitate concentration. The viscosity model is shown in the Appendix. Comparison of D a t a and Models Using "Bulk?' Properties Data for the three reactant concentrations and three coating compositions are plotted in Figures 6-8. The spreading models are represented in each figure. The solid linear region represents Cheever's equation, in which liquid properties are independent of time. The solid curve, departing from Cheever's model, is the undisturbed spreading model, with time varying fluid properties. This model assumes no change in effective capillary width during spreading. The break from the initial linear region oc-
curs when precipitation was observed to begin in bulk . dashed lines shown represent the capreactions ( t p ~ )The illary clogging models where they differ from the undisturbed flow model. The presence of more than one capillary clogging model for higher reactant concentrations is explained below. Comparison of data and models indicates that the undisturbed flow model does not adequately describe all the spreading data. The undisturbed flow model appears to fit best at 0.1 N reactant concentrations, except for 20% talc. The models may not be differentiable at this reactant concentration level, because of scatter in the data and the vertical shift from the model in individual sets of data due to differences in individual surfaces. In all other cases, the undisturbed flow model is not in good agreement with the data.
Table 111. Values of tp Observed for Bulk Polymerizations
Variation of the Capillary Clogging Parameters
An Alternate Model
The greatest discrepancy between the spreading data and models in Figures 6-8 occurs near the time at which precipitation was observed to occur in bulk reactions. Particularly a t high reactant concentrations, the measured drop diameter is less than that predicted by any of the models at t p ~when , departure from Cheever’s model was assumed to begin at t p ~ It. appears that clogging begins before the observed bulk onset of precipitation. In fact precipitation may occur earlier a t the Nylon-talc surface than it does in bulk. It is reasonable to expect that the presence of a seeding surface for polymer precipitation would not only increase the rate of polymer precipitation after time t p ~ but , it would also cause precipitation to occur a t a lower level of supersaturation. If the seeding effect were great enough, supersaturation might be eliminated. In either case, it is reasonable to assume that precipitation, and therefore capillary clogging, could occur earlier in the drop spreading system than the time observed in the bulk system. In order to determine the effect of varying the onset of precipitation, a variable t , was incorporated into the capillary clogging model. Both parameters tp and T~ were varied in order to determine if the fit between model and data could be improved. The predicted drop spreading curves were calculated from eq 4 and 5 . Some of the results obtained are plotted in Figures 6-8. As can be seen, decreasing tp shifts the predicted spreading curve to the lower left. along the solvent spreading curve which was calculated from Cheever’s model (eq 1). Decreasing T ~ which in effect blocks the capillaries faster, flattens the drop spreading curve. If T, is sufficiently small, the spreading curve approaches a limiting value before 30 min. By incorporating the two-parameter pore clogging models, the drop spreading data can be fit in nearly all cases. A summary of the parameters giving the best fit between models and data for all data, including those shown in Figures 6-8, is listed in Table 11. Incorporation of the two-parameter capillary clogging model as part of the spreading model does improve the fit of model to data. However, this is a shift from first principles, as employed in the derivation of Cheever’s model, to empiricism. At the present stage of development, three variables not apparent from theory have been included in the drop spreading model for complex fluids. They are the radial spreading exponent ( n )of eq 1, as well as t , and T ~ The radial spreading exponent can be measured from the spreading of simple liquids on a given porous surface. The other two’ capillary clogging parameters are not always experimentally accessible, however. At this point, the drop spreading model is no longer a predictive model, since in-
A third model was considered in which homogeneous fluid is wicked by capillary action from the gel formed at the center of spreading. Such a model assumes that gelation occurred some time after t p and that the capillary forces would not be sufficient to break the gel structure. Flow could continue, however, since capillary suction could remove liquid from the immobile gelled phase. The liquid would be a polyurea-saturated solution, however. As it spread and reacted, gelation would occur within it. This gelling-wicking-spreading process would continue until a balance between the capillary forces and the attractive forces of the gel for the solvent become equal. At that time, spreading would cease. This model may be more appropriate than the capillary clogging model at higher reactant concentrations, where the gelling tendency is greatest. A quantitative model of this situation requires mathematical definition of the gel formation and liquid extraction processes. One problem is that, as fluid is wicked from the gel, resistance to further removal would increase. That resistance must be incorporated into such a model. Also, the physical properties of the fluid actually spreading through the capillaries would not be known, since the reactant and polymer concentrations are not known. Because of these difficulties, the model was not developed. Lack of a model describing the gelling-wicking-spreading phenomena is not necessarily a major obstacle in understanding spreading of complicated fluid systems. Many fluids, consisting of solids suspended in a vehicle, will maintain fluidity even though changing viscosity or surface tension while spreading. For these systems, the capillary clogging model describes the spreading behavior.
Initial reactant concn, equiv/l. 0.1 0.15 0.2
(tJ‘R
300
180 120
(t,)ls
2 10 90 30
corporation of only bulk fluid properties and measurable surface parameters does not allow a reasonable estimate of the drop spreading radius as a function of time. It has been reduced to an empirical curve-fitting equation.
,
Summary
.
Experimental results have shown that a simple spreading relationship derived for pure .liquids also describes the flow of fluids with changing physical properties, if the spreading fluid remains homogeneous. When phase changes such as precipitation or gelation occur within the spreading fluid, the simple relationship is no longer valid. In the latter case, particle-capillary and gel-capillary interactions become increasingly important as the solid phase concentration increases. The spreading mechanism is now much more complex and depends upon the gelling and wicking interaction of the fluid-solid spreading system with the solid substrate. Appendix Drop Spreading Models. A. Homogeneous Fluid, Time-Dependent Fluid Properties. d is not a function of time. Also, for the system employed, the surface tension Ind. Eng. Chem.. Fundam., Vol. 14, No. 1, 1975 45
and contact angle did not vary. However, the general model for constant d is
The following viscosity approximations were used in calculating the undisturbed flow model curves in Figures 6-8 0 . 1 N: 17 = 13.7, tR 5 300 s e c 77 = 13.7 + (A2) 1 0 . 3 ( 1 - exp(-(t, - 180)/120)) 300 0.15N
