The Spreading Parameter

When a droplet is placed on a substrate (Fig. 1.6), there are two possible scenarios  

The parameter distinguishing between these two scenarios is the spreading parameter S, which measures the energy difference between the bare substrate (Tsa) and the substrate covered with a film of liquid (qsL + 7)  

If S 0, the energy of the bare substrate is greater, and the liquid will cover it (total wetting). If S 0, Young s formula shows that there exists an angle of contact [such that S = 7(cos e l)]j and the droplet will spread partially. 

At a certain temperature Tw, there may be a transition between partial and total wetting, i.e., e(2w) = 0. Tw is the temperature of the wetting transition. Another important value is 6e — which separates the so-called wetting liquids ( e tt/2) from the non-wetting liquids. This turns out to be important when studying the inhibition of powders (cf. Chap. 2) (although the temperature such that e( a) = 7t/2 does not correspond to a phase transition in the thermodynamic sense). 

Starting with the three surface tensions 7a, 7s, and 7ab, the spreading parameter can be determined  

If S is positive, wetting is total and the liquid spreads completely. Such is the case for a drop of PDMS deposited on the surface of water. 

If S is negative, wetting is partial. A drop can form a lens or a large puddle, depending on the amount of liquid involved. That is precisely what one observes when depositing oil on water or molten glass on liquid tin. 

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Microwave measurements have established with some high degree of probability that a small spread of relaxation times is exhibited by pure water. According to the Cole-Cole equation, the region between the static permittivity and the infinite frequency permittivity is spanned by the complex permittivity e, as defined in equation (13), where t is the principal relaxation time and a the spread parameter w = 2itv is the radial e = (Co - c )/tl + (j OT) -"] (13)... 

There is likewise very little doubt (> 99% certainty) that the spread of relaxation times is real. Moreover, as well as can be seen, the spread parameter a is temperature-independent. Its interpretation is still open to question. 

Note that, while the spread parameter crj is the same for all terms in the summation, the conditional parameter 0-2,0, can depend on a. Also, the functional form used for 6 1 need not be the same as that used for ( -g- 1 could use beta EQMOM, while 2 uses Gaussian EQMOM). Although the form in Eq. (3.134) is not as general as that in Eq. (3.124), we shall see that it leads to a direct method for moment inversion that is very similar to the one used in the CQMOM. The bivariate moments found from Eq. (3.134) have the form... 

In the present study, the mean droplet diameter is assumed to be 70.5 pm, the spread parameter is set at 2.09, and the corresponding minimum and maximum droplet diameters are taken as 10.0 and 138.0 pm, respectively. The corresponding spray mass flow rate is eqnal to 0.0139 kg/s (50 kg/h). The walls of drying chamber are made of 2 mm stainless steel, and the coefficient of heat transfer throngh the walls is set to zero thongh there is a perfect thermal insnlation of the chamber. Finally, the gage air pressnre in the ontlet pipe of the drying chamber is set to -100 Pa. 

Hence, the stabilization of polymer thin films against dewetting is of great importance to a number of technological applications. The likelihood that films will either wet or dewet solid surfaces is related to the value of the spreading parameter S ... 

Compared with the previous situation, liquid substrates exhibit fewer complications. Their surfaces are smooth and homogeneous, and there is no hysteresis effect. The surface and interfacial tensions of liquids (but not of solids ) can be measured, and the spreading parameter S is known. 

Once again, small drops adopt spherical cap shapes, and larger drops form fiat lense shapes. These floating lenses are subject to an Archimedean upthrust. Their thickness can be discovered by measuring their radii as a function of volume. The spreading parameter 5 = 7b — (7a + 7ab) can be deduced from the formula pe /2 = —S, where... 

In the case of styrene as monomer and hexadecane as model oil, the cohesion energy density of the polymer phase is closer to that of the oil, and therefore the structure of the final particles depends much more on those parameters which critically influence the interfacial tensions. A variety of different morphologies in the styrene-hexadecane system can be obtained by changing the spreading parameter. This was done by changing the monomer concentration and the type and amount of surfactant, as well as the initiator and the functional comonomer. 

This approximate expression can be used to estimate the spread parameter from the interaction parameter... 

It is evident that 6e can be defined only if the spreading parameter is negative. Oe increases when the liquid is non-wetting. 

Actually, the surface energy 750 in contact with air is not altogether sufficient to predict wettability. What is in fact needed is the sign of the spreading parameter S given by... 

For small drops of radius less than, the capillary forces are the only ones to come, into play. In accordance with Laplace s equation, their curvature must be constant. Therefore, a drop deposited on a horizontal surface takes on the shape of a spherical cap whose edges intersect the substrate at angle 0E Measuring that angle enables us to determine the spreading parameter (negative in the present case) through the expression 5 = 7 (cos - 1). 

We are now in a situation where the spreading parameter S is positive and the functional dependence of the energy P e) has the shape illustrated in Figure 4.4a. The construction of a simple tangent line reveals a threshold thickness Cc- Films with a thickness greater than Cc are stable. Films with a thickness smaller than Ce are either metastable or downright unstable. 

The only case where the above (macroscopic) discussion is significant is that in which the spreading parameter S is small. Since both P e) and ell(e) —> 0 when e —00, a small value of S implies a relatively thick film (a few tens of A) and the macroscopic approach is valid. For larger S values (small e) a detailed molecular description of the film would be required. 

Admittedly, the preceding calculation is unrealistic for small values of u because the expression for P u) diverges when ti —> 0. In actuality, P(u —> 0) remains finite and is equal to the spreading parameter 5 = —7 /2. In accordance with equation (4.35), the slope vanishes du/dx = 0) at the triple line. The correction is shown as a continuous line in Figure 4.8. In fairness, though, such distinctions affect regions of size a that are too small anyway for the calculation to be meaningful. 

We now consider the case when a liquid film A deposited on a substrate B (solid or liquid) is exposed not to air but to a deformable medium R, which can be either another liquid or a soft solid (an elastomer). The spreading parameter S is... 

A wedged-in drop will either spread or remain collected depending on the sign of the spreading parameter defined by... 

Young s relation is valid only in the immediate vicinity of the line since the shape of the drop will be determined by the elastic deformation of the elastomer. Together with the spreading parameter S and Young s modulus E of the rigid substrate, we can introduce a characteristic length ho, which we call the elastic length, defined by... 

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