Hydrodynamic and physicochemical aspects of spreading

Co~ids and Surfaces, 27 (1987) 43-55 Elsevier Science Publishers B-V., Amsterdam -

43 Printed in The Netherl~ds

Hydrodynamic and Physicochemical Aspects of Spreading* B.D. SUMM, V.S. YUSHCHENKO

and E.D. SHCHUKIN

Moscow Lo~onosou State University and the Institute of physical Chemistry of the Academy of Sciencesof the U.S.S.R., Moscow (U.S.S.R.) (Received 15 August 1986; accepted in final form 12 March 1987)

ABSTRACT A review is given of some experimental and theoretical results obtained by the authors and their coworkers concerning spreading. The following areas are covered: (1) the main experimental correlations, (2) the hydrodynamic theory, (3) spreading on rough surfaces, (4) the influence of physicochemical processes, and (5) the molecular dynamics of wetting.

INTRODUCTION

We began our experimental and theoretical investigations into spontaneous spreading in the early sixties at Moscow Lomonosov University. At that time this phenomenon had been little studied, but now a good deal of experimental data is available and some theories of the spreading process have been developed. These results are put to use in many modern technologies (soldering, sintering, oil recovery, working in non-gravity conditions, etc. ) , in agriculture and medicine (for example, in the production of contact lenses), in work at low temperatures, etc. The problems of spreading have been intensively investigated in many countries. Many scientists have worked in this field; among them are A.I. Bykhovsky, N.V. Churaev, B.V. Derjaguin, V.N. Eremenko, Yu.V. Goryunov, V.V. Khlynov, V.I. Kostikov, Yu.V. Naidich, V.A. Ogarev, S.I. Popel, L.M. Tcherbakov (U.S.S.R. ) , S.G. Mason (Canada), J.J. Bikerman, E. Dussan, E. Ruckenstein, L.E. Striven (U.S.A.), P. de Gennes (France), E. Wolfram (Hungary), J.F. Padday, T.D. Blake and J.M. Haynes (United Kingdom). The most important results and conclusions of their studies have been published in monographs and reviews. During the last 15-20 years this problem has also been considered at international conferences on surface and colloid chemistry *Dedicated to the memory of Professor E. Wolfram.

0166-6622/87/$03.50

0 1987 Elsevier Science Publishers B.V.

44

and surface active substances, as well as at some special conferences and symposia on wetting, adhesion and spreading. In this article we review experimental and theoretical results obtained on spreading at Moscow University (Dept. of Colloid Chemistry) and at the Institute of Physical Chemistry (Laboratory of Physicochemical Mechanics). We will cover the following aspects of spreading: 1. The main experimental correlations. 2. The hydrodynamic theory. 3. Spreading on rough surfaces. 4. The influence of physicochemical processes. 5. The molecular dynamics of wetting. Some of these results were obtained in collaboration with the Department of Colloid Chemistry of Lorand E&v& University, Budapest. For more than 20 years that Department was headed by Professor Ervin Wolfram. He often visited Moscow, delivered lectures at Lomonosov University and made reports at conferences. The authors would like to devote this review to the memory of Professor Wolfram. 1. THE MAIN EXPERIMENTAL

CORRELATIONS

Usually the spontaneous spreading of a small drop on a solid surface is studied. It is convenient to characterize the kinetics of spreading by means of the r(t) dependence; r being the radius of a wetted area, i.e. three-phase line (TPL) coordinate, and t being a period of time (the duration of spreading). So the first aim of our experiments was to determine the correlation between r and t. In the case of complete wetting, soon after initial contact with a solid surface, a drop changes into a thin round film which expands for quite a long period of time before it stops. For example, a 1 mm3 mercury drop spreads on solid zinc at room temperature for approximately 1 h. The kinetics of spreading, r(t) , was studied in detail for small mercury drops on solid metals: zinc, cadmium, tin, lead, indium, etc. [l-7]. The metal plates were placed in a weak acid solution to dissolve an oxide film. The kinetics of spreading depends strongly on the mass m of the mercury drop; two different types of r ( t ) function were observed: (1) For small drops ( m < 20 mg) , the experimental data (Fig. 1) can be satisfactorily fitted to the power function r=At”. The exponent n=0.22-0.26, and the coefficient A is proportional to m”.25. Besides two-dimensional (radial) spreading, a one-dimensional (linear) case was also investigated. In these experiments a long narrow “path” was prepared on a metal surface (the neighbouring parts were covered with unwetted dye). In this case it was found that the kinetics of spreading could also be described with the power function r =At”, where r is the distance between the drop centre

45

w

0

1

2

3

4

5

Fig. 1. Spreading of mercury drops on zinc. r (mm) is the radius of the time of spreading. Mercury mass m (mg) : (1) 1; (2 ) 5; (3) 10.

wetted area, t (s) is the

and the TPL. Here n=0.3-0.35 and A - (rn/~)‘.~, where a is the width of the path. (2 ) For large drops ( m > 100 mg) , two successive stages of spreading can be identified. They are both described by the function r=At”. At the first stage the exponent n=0.12-0.15, and the coefficient A - m”.35-o.40.Later, at the second stage, n= 0.22-0.26 and A - m”.25. The kinetics of spreading in the case of incomplete wetting was investigated in detail for liquids with relatively high viscosity. These experiments were performed with solutions of several metals in mercury, for example, with solutions of indium. The experimental data for spreading on cadmium and lead can be satisfactorily described with the equation r=At” (for small drops). The exponent n differs strongly from that in the case of complete wetting: the exponent n=O.lO-0.12 and coefficient Ah- rn’.‘. A similar correlation, r =A t’.l, can also be deduced for the spreading of other viscous liquids, for example, polydimethylsiloxane on Teflon [ 81, or glycerol and liquid polymers on Teflon, aluminium and mica [ 91. In summary, various cases of spreading may be described with the power function r=At”; the coefficient A is also a power function of drop mass m: A - m*. The experimental values of exponents n and z are recorded in Table 1. These power law correlations have been confirmed by many other experimental data obtained for various liquids (metals, oxides, polymers, organic substances, liquefied gases, etc. ) on various solids ( metals, graphite, polymers, semiconductors, etc.). These data are collected ir, several monographs and reviews [ 10-151. The power correlations have been explained by means of the hydrodynamic theory of spreading.

2. THE HYDRODYNAMIC

THEORY

The hydrodynamic theory of spontaneous spreading was proposed in 1963 Later it was developed on the basis of the Navier-Stokes equations

[ 3,6].

46 TABLE 1 Experimental

and theoretical

Experiment

Type of wetting

Complete small drops small drops large drops Incomplete small drops

values of exponents

n and .z for spreading

[l-7]

kinetics

Theory

[ 3,4,16,17]

n

z

n

2

(radial )

0.23-0.26 0.30-0.35 0.12-0.15

0.25 0.3 0.35-0.40

114 l/3 118

114 l/3 318

(radial)

0.10-0.12

0.3

l/10

3/10

(radial) (linear)

[ 12,16,17]. The main features and approximations of this theory are the following: (1) Soon after the initial contact of a drop with a solid surface the spreading velocity u = dr/dt becomes low enough for spreading to be considered as a quasistationary process. This condition is fulfilled when t >> ph*/q, where h is the average thickness of the liquid film. For the quasi-stationary process a driving force fd may be calculated by means of its thermodynamic determination. Namely fd = - dF/dx, where F is the free energy, and x is a generalized coordinate [ 181. For spontaneous spreading, F= F,+ Fg, where F, is the total surface energy of the three-phase system solid-drop-gas (or another liquid), and F, is the potential energy, x = r. ( 2 ) During the major part of the spreading the values of d*r/dt* are also so low that the inertia force is negligible compared with the viscous friction fv. The greater the radius of the wetted area r, the greater the viscous friction. Therefore it is possible to take into account only one resistance force, namely, viscous friction fv. This condition of non-inertial spreading is also deduced from the Navier-Stokes equations applied to the spreading process, namely: dr/dt -K qr/ph* [ 16,191. ( 3 ) Concurrent physicochemical processes (dissolution of a solid in the liquid phase, diffusion of a liquid in the solid phase, etc.) do not change (or change only very weakly) the surface and interfacial energies, viscosities and densities of liquids. Approximations (l)-(3) allow the kinetic equations of spontaneous spreading to be deduced for some particular cases. 2.1. Complete wetting A. Small drops, radial spreading

In this case only free surface -energy changes are important, so the driving force f,,=-dF,/dr. For calculating fd it is necessary to know the shape of the

41

spreading film. The simplest model is a thin cylinder V=m/p. For this approximation the thermodynamic driving force leads to the following equation: fd=2nr(as-as,-aL)

with constant determination

=2zrAa

volume of the (1)

where tss and trsL are specific free surface energies of the solid-gas and bL the surface tension of the liquid, and solid-liquid interfaces, Aa = gs - osL - oL, i.e. Harkins’ spreading coefficient, but with the opposite sign [20]. The next step is to calculate the force of viscous friction. For this purpose Newton’s law is used: fv = +S grad v

(2)

grad Vis calculated along the direction normal to the solid surface and approximately grad v=v/h, h=m/lrr2p is the average film thickness and S=nr2 is the friction area. From the balance fd = fy the following equation for radial (two-dimensional) spontaneous spreading is deduced: r= (Aam/qp)“4t”4

(3)

B. Small drops, linear spreading An analogous procedure was used for a drop spreading along a narrow “path” of constant width a. In this case fd = a (da), h = m/2rap, and S = ra. The kinetic equation of spreading is as follows: r= (Aam/2ap) ‘I3t l/3

(4)

C. Large drops, radial spreading For a large drop, the driving force is determined mainly by the decrease in the film potential energy Fg during its expansion and thinning; Fg= mgh/2, g being the gravitational constant. Therefore fd = -dF,/dr = m2g/zr3p. This driving force dominates over fd = -dFJdr if the film thickness h>> (2Aap/g) ‘j2. The viscous friction fv may be calculated with Eqn (2). In this case the balance fd = fv leads to the equation: r= (4m3g/x3p2rj)1’8t1’8 (the so-called “gravitational regime”). An identical spreading was deduced by Lopez et al. [ 211.

(5) equation

for this case of

2.2 Incomplete wetting, radial spreading During incomplete spreading a drop changes its shape. At the initial moment of contact ( t= 0) it is a sphere. Then at t> 0 the drop gradually transforms

48

into a spherical segment. The contact angle of the drop gradually diminishes from 180” to 13~,the equilibrium contact angle. The calculation of the spreading velocity in this case is more complicated than for the spreading of a film (in the case of complete wetting). For small drops, the driving force fd = -dF/dr. If a drop has a segmental form, this relation may easily be transformed into the well-known equation: f,=2ZFCTL(COSt&-COSO)

(6)

In order to describe the kinetics of spreading it is necessary to find the dependence fd( F) instead of fd( 8). The relation between F and cos 8 is very complicated. However, for acute angles (8 < 90” ) it is possible to use, with sufficient accuracy, the approximate equation 8 z 4 V/nr3 (where Vis the drop volume). Then Eqn (6) transforms into: fd=zrcrL(4V/7rr3)2[1-(7r710F3/4V)2]

(7)

The calculation of the viscous friction fv for a drop by means of Newton’s equation, Eqn (2)) is also rather complicated; for two reasons: in the first place, the thickness h here is not constant, diminishing from its maximum value at the centre to h=O (near TPL) ; and in the second place, the friction area S also varies from S = x? ( at the solid surface ) to S = 0 ( near the drop top ) . The viscous friction of a spreading drop may be calculated by an approximate method [ 31. The first step is to calculate the frictional force f* for a very narrow element dr, using Eqn (2). Then the integration fv = Jf *dr is made. This method leads to the following conclusion. The increase in the viscous friction in the spherical segment as compared with a round planar one may be taken into account by a dimensionless coefficient K z 10, which very slightly depends on drop radius r. Then Eqn (2) may be transformed to the following relation: fvzx(xr2q/V)dr/dt

(8)

From the balance between fd and f, it follows that dr/dt= (16V3aL/xn3ylrs)

[l-

(n00~3/4V)2]

(9)

Until r << r,, the integration of Eqn (9) results in [ 171: r=

[ (160/n3~) V3( aJr,y)t]

l/lo z V3’10(~L/~L)1’10t1’10

(10)

A similar relation was deduced by Tanner [ 221. Thus the hydrodynamic law describes the main period of spreading by means of the power equations r=At”. The kinetic coefficients A depend on surface ( oL, us, osL, 0,) and bulk (q,p) properties, and also on drop mass m (A N m*) . The calculated and experimental values of exponents n and z for mercury drops are summarized in Table 1. The hydrodynamic equations of spreading are also confirmed by many other experimental data covering different liquids and sol-

49

ids [ 11-141. The kinetics of spreading on solid surfaces immersed in an immiscible liquid (for example, glass/glycerol/paraffin oil) is also described with Eqn (10) [ 231. 3. SPREADING ON ROUGH SURFACES

The roughness of the solid surface may influence spontaneous spreading in two different ways. (1) The Wenzel-Derjaguin theory [ 24,251 predicts that a driving force fd should increase by K times in comparison with its value for a smooth surface ( see Eqns (1) and (5 ) ) ; K> 1 is then a roughness coefficient (factor). Therefore an increase in roughness (i.e. factor K) should induce an increase in the spreading velocity u. Such a correlation between u and K was observed for mercury drops spreading on solid metal surfaces of random roughness [ 11,121. (2) The roughness may also influence the resistance force f,.. In order to explain this connection let us consider a typical element of roughness - a groove with a constant cross section, for example a triangle. Along the groove the liquid should spread more quickly the greater the slope (Yof its sides is (i.e. the greater the roughness coefficient K= l/cos (Y). Conversely, a groove may delay or completely stop spontaneous spreading in the perpendicular direction. The condition for this was deduced by Gibbs [ 261: 8 -c 13,+ a, 8 being the contact angle at the leading edge of the groove. Such effects were observed by Mason and co-workers for various systems, including in the case of complete wetting (8,,= 0) [ 27,281, and also for alcohols and mercury drops on metal surfaces [ 29-311. Thus, the spreading on a rough surface is a complicated process. It is necessary to consider the scale and the shape of the various “hills and valleys” and also their disposition relative to the spreading direction. The most convenient way to study this problem is to carry out experiments on surfaces with regular relief, namely, on polished specimens with a single groove or with a system of parallel and crossing grooves. The experimental data available show that the spreading velocity is determined by the competition between the two tendencies mentioned above in relation to surface roughness [ 311. 4. THE INFLUENCE OF PHYSICOCHEMICAL

PROCESSES

During spreading, other physicochemical processes often take place which may affect the properties which determine the spreading velocity. The influence of the following processes has been studied in detail: (1) Diffusion of the liquid into the bulk of the solid. This process takes place if there is some solubility of the spreading liquid in the solid substrate. For example, mercury can dissolve in zinc, cadmium and many other metals. Diffusion results in a gradual decrease in drop mass m; therefore it causes

50

0

I

I

,

1

2

+m

Fig. 2. Radius of wetted area R (mm) for mercury drops of various masses m (mg) for spreading oncadmium (l),lead (2) andzinc (3) at2OT.

the rate of spreading to decrease [see Eqn ( 3 ) f . After the total mass m has diffused, spreading should stop. So the final radius R of the wetted area is partly determined by the competition between these two processes - spreading over the solid surface and diffusion into the bulk of the solid. This competition is especially important for complete wetting because spreading takes place over a long period of time. In this case the correlation between the final radius of spreading R and the drop mass m may be found by the following method [ 41. The loss of liquid dmd, which diffuses through a ring Znrdr ( r < R is the ring radius) is calculated by means of Fick’s law:

where c, is the liquid/solid solubility, D, is the liquid/solid diffusion coefficient, and td is the diffusion time. In this case td = tR - t,, t, is the time when the spreading liquid reaches a ring of radius r, and tR is the total period of spreading. The values oft, and tR may be calculated by means of the kinetic equation of spreading (3) (I Then the conditions Jdmd = m leads to the following equation 14): R= (h/qpcz

D~)~~am3~g

(11)

The power correlation RN m3j8 was confirmed by experimental data obtained for mercury drops spreading on certain metals [ 11,12,32,33] ; Fig. 2 illustrates this dependence. ( 2 ) Dissolution of grain boundaries. In many metallic systems a peculiar case of spreading is often observed. Its first stage is the rapid formation of a sessile drop on the solid surface ( BO> 0). Later, a round tarnished spot ( known as a “halo”) appears near the drop perimeter. For a long period of time (approximately 30 min for a 1 mg mercury drop on zinc) this halo slowly expands. In the end, it occupies a large area compared to the drop base:These

51

haloes were first observed by Tammann [ 341; many examples for metal systems are cited by Bondy [ 351. The halo spreading was investigated in detail for mercury drops on many solid metals [ 5,10,12]. The propagation of the halo front r(t) may be sufficiently correlated with the power function r=At0.5 (instead of rw t0.25 for “hydrodynamic” spreading). Increase of temperature T produces a considerable acceleration of halo propagation. Conversely, a drop mass m does not affect halo velocity. In order to explain the mechanism of halo formation and spreading, experiments with radioactive traces (for example, with *03Hg) and large grain specimens were performed [36]. The autoradiographs (for Zn-Hg systems) distinctly showed that mercury does not spread over the total solid surface; it spreads only along grain boundaries. These observations and kinetic data r=f(t) may be explained if a dissolution process (solid in liquid phase) is taken into account. According to the thermodynamic theory, grain boundaries should dissolve to a higher extent than grains. Therefore a network of microgrooves is formed on a polycrystalline surface in contact with a liquid phase. The equilibrium shape of the groove cross section is determined by Gibbs’ equation [ 261: where v, is the angle between the groove sides, and IJ~the cos (V/2 ) = Q/2%,, grain boundary free energy. If the thermodynamic condition sin (q/2 ) < cos ( 0,) (or v/2 < 90” - 8,) is fulfilled, a liquid may flow along such grooves; 0, is the equilibrium contact angle on a grain surface [ 1 l,l2 1. In this model the spreading velocity is determined by the process of grain boundary dissolution in a sessile drop. The kinetics of dissolution is usually limited by the diffusion of dissolved substance in the liquid phase. Using Fick’s law it is possible to deduce that the propagation of the halo perimeter should be described with the following equation: r= [20L(c,-c,)/p,]“2t”2

(12)

DL is a diffusion coefficient in liquid phase, pS a solid density, cg and c, are the solubilities of boundaries and grains ( cg> c,) [ 12,371. Equation (12) satisfactorily confirms experimental data (Fig. 3). It also explains the sharp temperature dependence of halo propagation velocity: &wexp(-l/T). 5. MOLECULAR DYNAMICS OF WETTING

The molecular mechanisms of studied [ 38,391 by the molecular jectories of the molecules in the method takes advantage of both

wetting and spreading processes have been dynamics method, i.e. by computing the trasystem under investigation [ 40,411. This experimental and theoretical approaches,

52

1

0 2

J

4

5

6

Fig.3.Kineticsofmercuryhalospreadingonzincat80°C (1),4O”C is the radius of the halo boundary, t (s) is the time of spreading.

(2) and20”C

(3).r(mm)

allowing us to reconstruct the real picture of the investigating process at the atomic level without any a priori hypothesis about its mechanism. The use of a two-dimensional system makes it possible to deal with very small systems, and the results can be easily visualized by displaying the molecular arrangement pictures at regular intervals. A movie film has been shot to demonstrate the motion of molecules during spreading and dewetting processes. The simulation of equilibrium properties of the micro-drop (Fig. 4c) showed that various macroscopic thermodynamic properties may be extrapolated up to the system of 19 moveable molecules. These parameters include the surface energies and the work of adhesion, equilibrium between the drop and monolayer, the conditions of complete wetting and so on. At the same time, corn-’ puter experiments showed that certain thermodynamic conceptions at molecular level are conventional, for instance the macroscopic contact angle is the result of extrapolation of the drop shape to the solid surface. Since there are only a few molecules near the line of the three-phase boundary this quantity becomes uncertain when the observation times are short, and should be determined by averaging the density profile over a very long time. Analysis of the spreading mechanism (Fig. 4a) showed that the transport of the liquid molecules onto the solid surface is realized by various processes that take place simultaneously. Some of these processes are: the cooperative motion of molecules - the flow of the liquid, surface diffusion at the drop surface to the three-phase line; evaporation/condensation mechanism; and the tearing of the molecules from the drop contour, i.e. diffusional transition to the adsorption layer. The molecular structure of the solid surface considerably influences the relation between the roles of the different mechanisms of spreading as compared with an idealized smooth suface: it slows the surface diffusion along the solid surface and makes the flow of the liquid more important. When the parameters of the system (intermolecular potentials) respond to

Fig. 4. Initial stats (top) and arrangement of molecules at various moments in the course of simulations. At the bottom are shown typical curves of variation of intermolecular bonds number for liquid-liquid (1) and solid-liquid ( 2 ) interactions. Time unit is F,, ( m/tLL) ‘I’; F, is an equilibrium distance between molecules (equal for liquid and solid), c is’ the energy parameter of the Lennard-Jones potential. ( 0 ) Liquid molecules (L) ; ( 0 ) molecules of surfactant (A) ; ( 0 ) molecules of a lyophilic solid ( Sl ) ; (0) molecules of a lyophobic solid. t LSlt cLSP, CAA? CLAY fAS2 are equal to 1,0.5,0.5,0.75,0.5 cLk For (a) - (d) see text.

incomplete wetting and the initial state is a monolayer (Fig. 4b), dewetting takes place - the molecules gather into a drop. As in the spreading process, the main mechanism here is the cooperative motion of liquid molecules; but weaker interaction between movable molecules and the solid surface facilitates the surface diffusion. The latter plays a more important role than for lyophilic surfaces. It is interesting that even in such a small system, instability of the monolayer may lead to its rupture into micro-drops which join together later. The dynamics of adsorption was studied on a system with molecules of a surface-active substance placed in the middle of the drop (Fig. 4d). The mol-

54

ecules of the surfactant go to the surface of the drop and improve wetting the system becomes more lyophilic due to the lowering of the surface tension of the liquid. Simultaneously the breaking of the drop is facilitated, and there is an equilibrium between spontaneous dispersion and coalescence of microdrops. The experiments performed showed that the molecular dynamics method makes it possible to observe the main physicochemical features of the wetting and spreading processes and opens up the prospect of carrying out quantitative analysis of the molecular mechanisms of various interactions at the interphase boundaries. At the same time it is obviously expedient to combine this method ( showing only short parts of phase trajectories) with the methods of statistical physics and the molecular theory of interphase layers.

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