Metrics for Reactive Wetting

in Complex Systems

METALS

For processes involving wetting between two materials, classically 

defined metrics such as surface or interface energy and contact angle 

are only a starting point for understanding the dynamic behavior of 

complex or nanometer-scaled systems that involve chemical reaction/

diffusion. The goal of this work is to have a quantitative model of 

reactive wetting. A good model should allow correct interpretation of 

dynamic measurements such as the wetting balance (a standard in 

solderability tests) and the meaning of contact angle measurement 

of systems not at equilibrium. Tools must also be developed to 

disseminate these results as codes usable by others. 

Objective

Impact and Customers

• A measurement need as stated in the IPC (Institute for Interconnecting and Packaging Electronic Circuits) report is 

the development of tests that properly capture how solder wets a printed wiring board, and to predict the service 

performance of the formed joint. 

• Models of this behavior should allow modifications to the widely used 

wetting balance test, with the goal of eliminating errors and extracting 

data more useful for solder joint design. 

• Additional areas of potential impact are found in nanofabrication with 

processes such as the controlled breakup of thin film deposits to form 

nanoscale droplets, or the growth of nanowires by the vapor-liquidsolid (VLS) method. These processes are encompassed by our theory, as 

the surfaces, interfaces and triple junctions of a three phase system are 

treated quite generally.

Approach

Materials Science and Engineering Laboratory

From fundamental physical considerations, we have derived a set of partial differential equations describing wetting and 

spreading. These equations are derived using a variational thermodynamic principle applied to a two component alloy 

system with three (vapor, liquid and solid) phases. The method naturally includes time dependent chemical interactions 

between substrate and liquid and also non-classical (diffuse interface) effects. The latter are important for applications 

where the system size approaches nanometer dimensions. 

Numerical solution of the equations will be performed for specific wetting 

geometries and parameter values. Results will be compared to predictions of 

simpler theories and experimental results. The importance of solute transport and 

its influence on the surface energy will be examined. A sensitivity analysis will 

determine those factors that dominate the wetting process. We will use the model 

to determine how to interpret the measurements of the wetting balance to best 

aid in the design of solder joints for fine pitch electronics. Subsequently, these 

tools will be applied to thin film breakup, heterogeneous nucleation on shaped 

particles, and VLS nanowire growth.

855-13-31b
Materials Science and Engineering Laboratory

We have implemented a numerical 

solution scheme to solve these equations 

and are examining the predictions. Four 

coupled partial differential equations 

comprise the model: a mass balance 

equation, a modified Navier-Stokes 

equation for the velocity of the matter in 

the various phases, a modified diffusion 

equation for the concentration field, and 

an equation to treat the dynamics of 

the solid-fluid interfaces (phase field). 

Solving this system of coupled equations 

accurately with appropriate parameters 

requires sophisticated numerical solution 

techniques and long simulation times. Our 

approach thus far has yielded a complete 

solution of the equations in two dimensions 

with a slightly nonrealistic parameter set to 

enable the solution of the equations in a 

practical time frame. 

An adaptive mesh is being used to solve 

the differential equations. A fine mesh 

is needed near the three interfaces 

between the liquid, solid and vapor for 

the spreading drop problem in order to 

have the numerical accuracy necessary to 

maintain correct changes in concentration 

across interfaces. The mesh moves (adapts) 

as the droplet spreads and the solid is 

partially dissolved. Only a coarse mesh is 

needed in the interior regions of the liquid, 

solid and vapor phases because gradients 

are less severe there.

A plot of contact angle (theta) versus triple 

junction speed (capillary number, Ca*) 

shows a large contact angle at high speed 

that decays to the equilibrium contact angle 

as the triple junction slows down to zero 

when spreading is complete. The general 

shape of the curve is known from fluid 

mechanics analysis and measurements 

for nonreacting cases (solid curve). In 

the model, this limit has been recovered 

numerically by setting solute diffusion and 

interface mobility parameters to zero. This 

confirms the validity of the model. For 

non-zero values of these parameters, the 

contact angle versus triple line speed curve 

is altered. The wetting is generally faster 

than the fluid mechanics case when phase 

change and diffusion are allowed.

Fine scale details of the concentrations 

near the moving triple line are also being 

examined. The changing concentrations 

near the triple line change the local values 

of the surface energies and affect the 

contact angle. The top figure (large contact 

angle and high speed) at the early stage 

of spreading shows a relatively uniform 

liquid concentration with an altered vapor 

concentration in front of the moving liquid 

droplet. At later stages (bottom figure), the 

concentration gradients exist mostly in the 

liquid and solid.

Accomplishments

James Warren 

Walter Villanueva 

Daniel Wheeler 

William J. Boettinger 

(Metallurgy Division)

(301) 975-6160 

william.boettinger@nist.gov 

www.nist.gov/metallurgy

Learn More Publications

Moon KW, Boettinger WJ, Williams ME, Josell D, Murray B, Carter WC and Handwerker 

CA Dynamic Aspects of Wetting Balance Tests J. Electronic Packaging, 118: 174 (1996)

Warren JA, Boettinger WJ and Roosen AR Modeling Reactive Wetting Acta Materialia, 96: 

3247 (1998)

Granasy L, Pusztai T, Saylor D and Warren JA Phase-field Theory of Heterogeneous 

Crystal Nucleation Physical Review Letters, 98 (3): 035703 (2007)

Warren JA, Pusztai T, Környei L and Gránásy L Phase Field Approach to Heterogeneous 

Crystal Nucleation in Alloys Phys. Rev. B, 79: 014204 (2009)

Adaptive mesh used for numerical computation.

Contact angle (theta) vs. triple junction 

speed (Ca*).

Enlarged view of concentration field near the 

moving triple junction at three times.