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Langmuir 1998, 14, 528-532
Strange Spreading Behavior of Tricresyl Phosphate Martin E. R. Shanahan,*,† Marie-Christine Houzelle,‡ and Alain Carre´*,‡ CNRS, ENSMP, Centre des Mate´ riaux P. M. Fourt, BP 87, 91003 Evry Ce´ dex, France, and Corning S.A., Fontainebleau Research Center, 7 bis, avenue de Valvins, 77210 Avon, France Received February 25, 1997. In Final Form: October 6, 1997 The spreading of a sessile drop can be explained by a dynamic energy balance in which released excess capillary energy causes triple-line motion while wetting front speed is moderated by viscous dissipation within the liquid. This energy balance leads to a well-known relationship between dynamic and equilibrium contact angles and spreading speed. Experiments conducted with a silicone oil, poly(dimethylsiloxane), confirmed the validity of this equation. Similar experiments of spreading of tricresyl phosphate (TCP) on the same three substrates, glass, Halar (polyethylene-chlorotrifluoroethylene), and polypropylene, led to anomalous behavior: among other observations, we noted that the (apparent) spreading force needed for a given spreading speed was too high. Here, we suggest a tentative theory involving molecular orientation or conformation of the asymmetrical liquid molecules after they have been “laid down” on the solid substrates during spreading. As TCP molecules orientate near the triple line, the liquid/solid interfacial tension and therefore the local value of equilibrium contact angle are modified. The theory adequately explains several experimental results but leaves some unanswered questions.
Introduction The spreading of liquids on solid surfaces invokes some strange behavior and phenomena, not the least of which is the still much discussed contravention of the usual Navier-Stokes no-slip condition1 at the liquid/solid interface in the immediate proximity of the wetting front.2-4 Nevertheless the basic physics of a classical liquid wetting (or dewetting) an ideal solid substrate in the absence of significant body forces (gravity) can be expressed quite simply as a dynamic energy balance.3 If the liquid is not at equilibrium, excess capillary energy is released, leading to a driving force for spreading. As the liquid shears during motion, energy dissipation occurs and moderates the wetting (or dewetting) speed. The rates of release of capillary free energy and viscous dissipation are equal, and in the specific case of wetting, spreading speed decreases as the contact angle, θ, tends toward the equilibrium value, θ0. Basic equations will be summarized below. In this elegantly simple description, both solid and liquid are assumed to be ideal, the former representing a flat, smooth, rigid, homogeneous, and isotropic surface and the latter being a pure (to eliminate surface tension gradient effects), Newtonian material of low molecular weight (polymer melts can lead to slip phenomena3). In the present study, we have considered the spreading of a commonly used liquid in dynamic wetting experiments, a silicone oil, poly(dimethylsiloxane) (PDMS),5 and of one considered to be essentially apolar and often used in solid surface characterization,6 tricresyl phosphate (TCP). The spreading behavior of these two liquids has been studied on three rather different solids: glass, Halar (polyethylene-chlorotrifluoroethylene, Ausimont), and polypropylene (Goodfellow). It is shown how indeed PDMS may * Corresponding authors. † CNRS. ‡ Fontainebleau Research Center. (1) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics: Course of Theoretical Physics, 2nd ed.; Pergamon: Oxford, 1987; Vol. 6, p 47. (2) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. (3) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (4) Thompson, P.; Robbins, M. O. Phys. World 1990, November, 35. (5) Carles, P.; Cazabat, A. M. Prog. Colloid Polym. Sci. 1990, 82, 76. (6) Fowkes, F. M. Ind. Eng. Chem. 1964, 56, 40.
be considered to be an ideal liquid spreading on the substrates in question, whereas TCP behaves somewhat strangely. We propose a theory which adequately explains the unexpected behavior, although leaving unanswered questions. Summary of Basic Theory3 Consider a unit length of triple-line solid/liquid/vapor for which the contact angle, θ(t), is a function of time, t, and greater than the equilibrium value, θ0, for the three given phases; this corresponds to spreading at speed U as shown schematically in Figure 1. The spreading force is given by
FS ) γSV - γSL - γ cosθ(t) ) γ[cos θ0 - cos θ(t)]
(1)
where γSV, γSL, and γ represent respectively the surface (interfacial) tensions at the solid/vapor, solid/liquid, and liquid/vapor interfaces. The second equality follows directly from Young’s equation. As spreading occurs at speed U, work is supplied at a rate FSU. Assuming the contact angle to be sufficiently small (θ e 1 rad), the flow field within the liquid may be taken to be a parabolic distribution, in the lubrication approximation, with boundary conditions of no slip at the solid surface and no shear at the free liquid surface.3 This leads to a distribution of speed, v(z), as a function of distance from the solid surface, z, given by
v(z) )
3U (2hz - z2) 2h2
(2)
where h is the height of the free liquid surface above the solid. Realizing that viscous dissipation (per unit volume) takes the form η(∂v/∂z)2 where η is viscosity, it may be shown that the overall dissipation per unit length of triple line and per unit time is given by
TS˙ ≈
3ηU2 l θ(t)
(3)
where l is the logarithm of the ratio of a macroscopic
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line. Regression analysis of the results for PDMS leads to the relation
log{θ(t)[cos θ0 - cos θ(t)]} ) 1.00 log
Figure 1. Schematic representation of contact line region in motion (spreading).
distance, of the order of contact radius in the case of a sessile drop, and a microscopic cutoff distance near the triple line. The cutoff is introduced to prevent divergence at the triple line and corresponds (probably) to breakdown of the no-slip conditions in the immediate vicinity of the wetting front, as evoked in the Introduction.4 Despite its interest, discussion of this matter would represent a digression here; we shall treat l as a constant. Typically it has a value on the order of 12.7 Equating dissipation, TS˙ , to work supplied, FSU, we obtain
θ(t)[cos θ0 - cos θ(t)] ≈
3ηlU 3lU ≡ γ U*
(4)
where U* represents a characteristic spreading speed for the liquid, γ/η. Some approximations are necessary to obtain a tractable expression for dissipation, eq 3, but none for the driving force, eq 1. This explains the mixture of angle and trigonometric functions on the left-hand side of eq 4. Although not entirely consistent, we have preferred to use this form since some of the angles treated below approach 1 rad or less. Experimental Section Flat, horizontal surfaces of glass, Halar, and polypropylene (PP) of surface roughness Ra lower than 0.03 µm were used for observing the spreading of 0.2 µL drops of a poly(dimethylsiloxane) (Rhodorsil 47V 1000, Rhoˆne-Poulenc) and tricresyl phosphate (mixture of isomers, Aldrich Chemicals) at ambient temperature (20 ( 2 °C) and relative humidity (RH ) 50%). The possible impact of the chemical purity of TCP will be discussed in the last section (Discussion and Conclusion). Data concerning the liquids’ viscosity, η, surface tension, γ, density, F, and the equilibrium contact angle, θ0, on the various solids are presented in Table 1. All contact angle measurements were obtained by using a contact angle goniometer (Rame´-Hart A-100) equipped with a video camera connected to a video recorder and to a printer. The contact angles, θ(t), were measured as a function of time after deposition, t. It was possible to analyze up to 24 frames/s. The speed of motion of the triple line, U, was obtained from the variation of drop radius, r, as a function of time, t, i.e., U ) dr/dt.
Results and Interpretation 1. Comparison of Behavior of PDMS and TCP. Figure 2 shows spreading results of both PDMS and TCP drops on the three solid surfaces studied using a representation in accordance with eq 4, viz., θ(t)[cos θ0 - cos θ(t)] vs U/U*, both coordinates being on logarithmic scales. We notice immediately that for a given liquid, using a normalized abscissa, we obtain results for the three different solid substrates which lie on a unique straight (7) Ondarc¸ uhu, T.; Veyssie´, M. J. Phys. II 1991, 1, 75.
{U*U } + 1.45
(5)
with a correlation coefficient of r ) 0.987. Thus experimental results are in excellent agreement with the prediction of eq 4. Moreover, we have log (3l) ) 1.45 corresponding to l ) 9.4. This value is perhaps a little lower than may be expected,7 but l is sensitive to molecular size. We may therefore conclude that the classic theory3 works very satisfactorily in the case of this model liquid. Although the results for spreading of TCP on the three surfaces also lie satisfactorily on a straight line, regression analysis leads to
log{θ(t)[cos θ0 - cos θ(t)]} ) 0.77 log
{U*U } + 1.34
(6)
with a correlation coefficient of 0.981. In other words, the left-hand side of eq 4 scales approximately with U0.8. In addition, despite normalization of the coordinates, the results lie very definitely above those for PDMS; that is, a greater than expected (apparent) spreading force is required for a given spreading speed. Clearly something is awry, and we shall tentatively suggest an explanation for this strange behavior. The molecule of TCP has the formula (CH3(C6H4)O)3PO and therefore an intrinsic asymmetry. Although the molecule has an isolated oxygen atom attached to the phosphorus, it is commonly assumed that its apolar nature is related to the fact that orientation or conformation occurs at the liquid surface (interface) in such a way that the essentially apolar organic groups are exposed. This will (presumably) be the situation at equilibrium. However, during spreading, TCP molecules are “laid down” on the solid substrate in what must initially be a fairly random manner. Once on the solid surface, the liquid molecules may reorientate themselves, aided by thermal agitation, with the aim of finding the most suitable orientation or conformation, compatible with a local free energy minimum. The kinetics of this operation will govern the local interfacial free energy, γSL, at a given time following initial liquid/solid contact. It is now possible to imagine, at least qualitatively, how, as liquid spreading continues at a variable rate, the local γSL may change, thus modifying the capillary “motor” for spreading, viz., eq 1. Let us attempt to quantify the concept. 2. Tentative Theory of Anomalous Spreading of TCP. Computational molecular modeling of possible conformations of TCP demonstrates that the molecule can adopt in fact several conformations of very similar free energies. In this study, we will consider only two different model conformations in order to simplify the mathematical analysis. These two conformations are schematically represented in Figure 3. Consider Figure 3a, which represents the TCP molecule in a simplistic way with three organic branches and the isolated oxygen. We shall assume that the effective surface free energy or tension of the side presenting organic groups is γE and that of the other extremity γO and that γE < γO. This is in fact a simple scheme to express that the surface free energy of TCP depends on its conformation (or orientation). Similarly we take it that γSE < γSO, where γSE and γSO are the interfacial free energies (tensions)
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Table 1. Data Concerning the Two Spreading Liquids and Their Contact Angles at Equilibrium on the Three Solid Substrates at 20 °C liquid
viscosity, η (Pa‚s)
surface tension, γ (mN‚m-1)
density, F (kg‚dm-3)
poly(dimethylsiloxane) (PDMS) tricresyl phosphate (TCP)
0.971 0.07
21.2 40.9
0.971 1.143
a
equilibrium contact angle,a θ0(0) G H PP 0 18
0 35
0 45
G ) glass, H ) Halar, and PP ) polypropylene.
angles for the two extreme cases:
cos θI )
γSV - [(γSO + γSE)/2] γ
(8)
γSV - γSE γ
(9)
cos θE )
Figure 2. Results of spreading of drops of PDMS and TCP on glass, Halar, and PP surfaces expressed as θ(t)[cos θ0 - cos θ(t)] vs normalized speed, U/U*.
Clearly from eqs 8 and 9 and the preceding reasoning, we have cos θI < cos θE, or θI > θE. Both θI and θE represent equilibrium contact angles but for different conformations or configurations of the liquid molecules; however θI is a hypothetical equilibrium due to ensuing orientation. Let us assume that molecules contributing γSE/2 in eq 7 remain in their equilibrium state and that those giving γSO/2 turn over obeying first-order kinetics (this will not be strictly true due to thermal agitation, but should be a fair approximation). For nO TCP molecules per unit interfacial area, we have n turned to the optimal conformation; then
dn 1 ) (nO - n) dt˜ τ
(10)
where τ is an effective time constant. At ˜t ) 0, n ) nO/2 and so
n ) nO[1 - exp(-t˜/τ)/2]
(11)
Consideration of the above now allows us to write for the interfacial free energy, γSL(t˜), as a function of time, ˜t, after solid/liquid contact,
γSL(t˜) ) γSE + (γSO - γSE) exp(-t˜/τ)/2
(12)
or, alternatively, Figure 3. Schematic representation of contact line region and hypothesized orientation process of TCP molecules. 4 is the distance over which the molecular interactions are averaged leading to local γSL ()γ j SL), and therefore contact angle, and δ is the distance at which “most” molecules have “flipped over”.
between the solid and, respectively, sides E and O of the TCP molecule. (It may of course transpire that γO < γE and γSO < γSE, but the argument is unchanged; we assume the first possibility for definiteness.) When the liquid is first “laid down”, (local) time ˜t ) 0 and the molecules are randomly distributed. Therefore, assuming additivity, the interfacial free energy immediately at the triple line is given by
γSL(t˜ ) 0) )
γSO + γSE 2
(7)
For ˜t ) ∞, γSL(∞) ) γSE, but for 0 < ˜t < ∞, γSL(t˜) will be somewhere between the two; nearer to the value given by expression 7 for small ˜t values, and nearer to γSE for large ˜t values, the order of magnitude of ˜t remaining to be determined. Using Young’s equation, we define contact
cos θ0(t˜) )
γSV - γSL(t˜) ) cos θ0(∞) + [cos θ0(0) γ cos θ0(∞)] exp(-t˜/τ) (13)
where we take θ0(t˜) to be a theoretical equilibrium contact angle for time ˜t after the TCP contacts the solid. A similar argument has been previously used to explain effects of swelling in the kinetics of wetting on rubber.8 Figure 3b is a schematic representation of the local TCP molecules near the spreading front of a drop. Those very near the triple line are randomly distributed, having just been “laid down”, whereas those toward the right have been in contact for some time, increasing with distance from the triple line, and as a consequence, the fraction in their optimal conformation (orientation) increases. We shall assume that the orientation process is rapid such that we may define a distance, δ, from the triple line (see Figure 3b) for which “most” molecules have “flipped over”. At δ, the local ˜t ≈ τ. The contact angle will depend on the average value of γSL over some (small) distance from the triple line, ∆, but γSL is a function of distance, x, due to (8) Shanahan, M. E. R.; Carre´, A. J. Adhes. 1996, 57, 179.
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Langmuir, Vol. 14, No. 2, 1998 531
variable local contact time, ˜t (which increases with x in the figure). Distance ∆ will be a constant, presently unknown but on the order of the range of van der Waals bonds, say 10-50 nm, whereas δ depends on spreading speed, U. We shall therefore make a simple estimate of the j SL from average “local” value of γSL ) γ
∫0
∆
γ j SL ≈ ∆-1
γSL(x) dx
(14)
where
γSL(x) ≈ γSE + (γSO - γSE) exp(-x/(Uτ))/2
Table 2. Values of L ) (γSO - γSE)/(2γ) and v ) ∆/τ Calculated for TCP Spreading on the Three Solids Studieda solidb
L
v (m‚s-1)
τ (s)
G H PP
6.0 × 10-2 7.8 × 10-2 8.2 × 10-2
8.5 × 10-5 2.4 × 10-4 1.9 × 10-4
1.2 × 10-4 4 × 10-5 5 × 10-5
av
(7.3 ( 1.2) × 10-2
(1.7 ( 0.8) × 10-4
(7 ( 4) × 10-5
a The time constant, τ, for orientation of TCP molecules is calculated on the assumption that ∆ ≈ 10-8 m. b G ) glass, H ) Halar, and PP ) polypropylene.
(15)
These simplified expressions neglect any weighting due to distance x in the contribution to γ j SL and assume any variation in U over the spreading distance ∆ to be small, as approximations. We thus obtain
γ j SL(U) ≈ γSE +
-∆ Uτ (γ - γSE) 1 - exp 2∆ SO Uτ
(
( ))
(16)
Replacing γSL(t˜) in eq 13 by expression 16 leads to
cos θ0(U) ) cos θ0(U ) 0) -
Uτ (γ - γSE) × 2γ∆ SO -∆ 1 - exp Uτ
(
( )) (17)
where θ0 is now considered to be dependent on spreading speed, U, rather than contact time, ˜t, directly. Returning to eq 4, we can see that if this orientation effect exists, cos θ0 must be replaced by the spreading rate dependent expression 17. The resulting relationship is
cos θ0(0) - cos θ(U) -
LU v 3lU 1 - exp ≈ v U θ(U)U*
(
( ))
(18) where the 0 of cos θ0(0) refers to 0 spreading rate, or infinite time,
L ) (γSO - γSE)/(2γ) and v ) ∆/τ Equation 18 is the central result of this study and predicts how the “target” equilibrium contact angle may change due to molecular orientation (conformation) of the liquid, leading to modified spreading. Unfortunately, its application is not straightforward, but we shall endeavor to analyze the TCP results of Figure 2 using the following technique. We commence by rewriting eq 18 in the form
cos θ(U) U 3l + ) U* cos θ0(0) θ(U) cos θ0(0) v L U 1 - exp 1U v cos θ0(0)
(
( )) (19)
Taking l ) 12, a value generally accepted as reasonable,7 we are in a position to calculate the value of the left-hand side of eq 19 for a given pair of values (U, θ(U)), the remaining constants being available from Table 1. This is done for two pairs of values (U, θ(U)), and we thus arrive at two (nonlinear) simultaneous equations with θ and v as unknowns (on the right-hand side). In principle, the pairs of values should correspond to extreme values of spreading speed, but in practice, there is much less precision for very low speeds (drops near equilibrium),
Figure 4. Spreading results for TCP on glass: 9 experimental results of Figure 2 and ] predicted values from eq 20. The regression lines for both TCP and, for comparison, PDMS for all results are given.
and therefore, our lower speed pairs were chosen from somewhere near the middle of the (logarithmic) range studied. Values of L and v were calculated for TCP on the three solids in this way: results are summarized in Table 2, together with estimated values of the time constant, τ, calculated on the assumption that ∆ ≈ 10-8 m. After the values of the parameters L and v (and l) are estimated, it is possible from experimental values of U and θ(U) to construct a graph with the same coordinates as Figure 2 (but with U rather than t dependence), using a rearranged form of eq 18 to predict the values of the ordinate:
θ(U)[cos θ0(0) - cos θ(U)] ≈ 3lU LUθ(U) -v + 1 - exp U* v U
(
( )) (20)
Figures 4-6 show both the original, experimentally obtained data for spreading on the three solids together with their calculated, common regression line and the equivalent values expected from eq 20. The pairs of values of U and θ(U) corresponding to points A and B were those used to calculate L and v. Discussion and Conclusion It can be seen from Figures 4-6 that there is adequate agreement between the values of the ordinate calculated from eq 20 and those obtained more directly using experimental values alone. Regression lines for each individual set of results, i.e., for TCP on each solid, could have been calculated, but since Figure 2 strongly suggests that, once normalized, all belong to a common family, we have not endeavored to do this. Even so, the agreement between the common regression line and the values calculated from eq 20 is satisfactory, and there is a clear differentiation between these results and those for PDMS
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final value, given by γSE after orientation:
∆γSL )
Figure 5. As for Figure 4, but the substrate is Halar: g experimental results of Figure 2 and ] predicted values from eq 20. The regression lines for both TCP and, for comparison, PDMS for all results are given.
γSO - γSE ) γL ≈ (3 ( 0.5) mN‚m-1 2
This value seems plausible and may represent something like 10% of the overall value of interfacial tension. Concerning the values of v and τ in Table 2, this aspect is open to discussion. The characteristic speed, v, depends on both the effective bond range, ∆, and the time constant for orientation, τ. Neither is known with any degree of certainty! Assuming our estimate of ∆ at 10-8 m to be reasonable, a resulting time constant on the order of 10-4 s might seem rather large for a molecular orientation process. However, the nature of the process is not really understood; it will probably involve some sort of dynamic phenomenon invoking thermal agitation and, perhaps, local flow patterns. In addition, the development of eq 15 et seq. is somewhat simplified since dependence on the distance of interaction is neglected. This may also have some influence on the values obtained. The series of experiments extensively analyzed in this paper concern a mixture of ortho, meta, and para TCP isomers (Aldrich Chemicals). Of course, the question of potential influence of the chemical composition of TCP can be raised. Nevertheless, other series of experiments were conducted with pure o-TCP (99%, Acros Organics) spreading on glass. For this isomer, eq 6 was found slightly modified and satisfying
log{θ(t)[cos θ0 - cos θ(t)]} ) 0.86 log
{U*U } + 1.23 (21)
Figure 6. As for Figure 4, but the substrate is polypropylene: f experimental results of Figure 2 and ] predicted values from eq 20. The regression lines for both TCP and, for comparison, PDMS for all results are given.
(represented simply by their regression line in Figures 4-6). There is some tendency for the calculated results to drop below those obtained experimentally for lower values of U/U*, but as mentioned above, in this range of speeds near final drop equilibrium, experimental values are of reduced precision anyway (values of U are typically on the order of 10-5 m‚s-1). This is the presently preferred explanation to account for the experimental results which do not fit de Gennes’ theory. Assuming the tentative theory to be correct, we may calculate from the values of L the variation in effective interfacial free energy between TCP and the three solids depending on its molecular orientation. Since the values of L for the three substrates are similar with an average value of (7.3 ( 1.2) × 10-2, we may calculate the difference, ∆γSL, between initial interfacial tension, given by (γSO + γSE)/2 for a random liquid molecular distribution, and the
with a coefficient of correlation of 0.985. Again, the slope of the linear relationship (21) is clearly smaller than 1. Moreover, similar experiments were also conducted with purified glycerol trioleate (Prolabo). As for TCP and o-TCP a slope smaller than 1 (0.89) was obtained when de Gennes’ theory was applied. All these results indicate clearly that viscosity is not the only parameter to take into consideration to describe the kinetics of liquid spreading. It is believed that a secondary phenomenon, namely molecular orientation, reduces the speed of wetting for liquids constituted of asymmetric molecules. This study leaves certain questions unanswered, and clearly further work in the field is necessary to obtain a greater understanding of the anomalous spreading behavior of liquids constituted of asymmetric molecules on solid substrates. Nevertheless, a fairly simple theory has been developed along the lines of molecular orientation of the liquid leading to variable equilibrium contact angles, and this seems to explain the observation adequately, if not perfectly. LA970206M
