# Rheological characterization of non-Brownian suspensions based on structure kinetics

## Abstract

### Purpose

The purpose of the paper is to develop a methodology to characterize the rheological behaviour of macroscopic non-Brownian suspensions, like solder paste, based on microstructural evolution.

### Design/methodology/approach

A structure-based kinetics model, whose parameters are derived analytically based on assumptions valid for any macroscopic suspension, is developed to describe the rheological behaviour of a given fluid. The values of the parameters are then determined based on experiments conducted at a constant shear rate. The parameter values, obtained from the model, are then adjusted using an optimization algorithm using the mean deviation from experiments as the cost function to replicate the measured rheology. A commercially available solder paste is used as the test fluid for the proposed method.

### Findings

The initial parameter values obtained through the analytical model indicates a structural breakdown that is much slower than observations. But optimizing the parameter values, especially the ones associated with the structural breakdown, replicates the thixotropic behaviour of the solder paste reasonably well, but it fails to capture the structure build-up during the three interval thixotropy test.

### Research limitations/implications

The structural kinetics model tends to under-predict the structure build-up rate.

### Practical implications

This study details a more realistic prediction of the rheological behaviour of macroscopic suspensions like solder paste, thermal interface materials and other functional materials. The proposed model can be used to characterize different solder pastes and other functional fluids based on the structure build-up and breakdown rates. The model can also be used as the viscosity definitions in numerical simulations instead of simpler models like Carreau–Yasuda and cross-viscosity models.

### Originality/value

The rheological description of the solder paste is critical in determining its validity for a given application. The methodology described in the paper provides a better description of thixotropy without relying on the existing rheological measurements or the behaviour predicted by a standard power-law model. The proposed model can also provide transient viscosity predictions when shear rates vary in time.

## Keywords

#### Citation

Vachaparambil, K., Mårtensson, G. and Essén, L. (2018), "Rheological characterization of non-Brownian suspensions based on structure kinetics", *Soldering & Surface Mount Technology*, Vol. 30 No. 1, pp. 57-64. https://doi.org/10.1108/SSMT-08-2017-0021

### Publisher

:Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited

## Introduction

Thixotropy is defined as “the gradual decrease of viscosity under shear stress, followed by a gradual recovery of structure when the stress is removed” (Barnes *et al.*, 1989). In thixotropic fluids, the spatial arrangement of constituent particles, commonly referred to as the microstructure (Vermant and Solomon, 2005), affects macroscopic properties like viscosity. The aqueous polymeric dispersions used in pharmaceutical industries, whose microstructure is based on the interactions between the constituent polymer chains like carbomer microparticles (Islam *et al.*, 2004), is one of the many examples of thixotropy observed in nature and industries.

The microstructure observed at any instant in time is a result of an equilibrium between the relevant forces such as Brownian motion, interparticle forces and hydrodynamic forces. The microstructure can also be understood in terms of the association between particles, which is dependent on the aforementioned forces. At equilibrium, either the Brownian motion or the interparticle forces, or a combination of both, dictate the particle distribution. Owing to the extensive interparticle associations when the fluid is at rest, the viscosity of the fluid is higher. These associations between particles are broken when force is applied, causing the viscosity to reduce. The transient microstructural changes are evident in the formation of flocs when no stresses are applied on the fluid and the destruction of flocs when force is applied, resulting in a lower viscosity (Barnes, 1997). The rheological behaviour of suspensions, where Brownian motion is the dominant physical mechanism, is investigated by Willey and Macosko (1978). The influence of Brownian motion can be expressed using the Péclet number, *Pe* = 6πηγ̇*r*^{3}/(*kT*), where *r* is the radius of the particle, γ̇ is the applied shear rate, *η* is the viscosity of the fluid and *kT* indicates the thermal energy associated with the Brownian motion. The Péclet number expresses the ratio between the shear forces in the flow and the Brownian motion of the particles. The suspensions where the Brownian motion can be neglected, when *Pe* ≪ 1, are called macroscopic suspensions. An example of a macroscopic suspension is mentioned by Völtz *et al.* (2002), where an experimental investigation into the thixotropic behaviour of a Newtonian carrier fluid with a high packing density of glass beads is carried out. Blood, which consists of non-Brownian red blood cells (Higgins *et al.*, 2009), exhibits thixotropy when shear rates applied are below 1 s^{−1} (Huang and Horng, 1995), and is another example of macroscopic suspension in nature.

Because the microstructure plays an important role in the rheological behaviour of the thixotropic fluid, methods with varying levels of complexity have been used to aid in the understanding of its behaviour. These methods include microstructural models, viscosity theories and structural kinetics models. The microstructural model describes physical mechanisms at the particle level that govern the microstructure build-up and breakdown (Potanin *et al.*, 1995). This approach is based on the ability to extensively describe the mechanisms that cause the macroscopic changes in fluid properties like viscosity (Mewis and Wagner, 2009). The viscosity theories uses measurable parameters, like yield stress, to circumvent the need to use the parameters to describe the evolution of the microstructure (Nguyen and Boger, 1985). A more detailed explanation about this technique can be found in Barnes’ (1997) study.

An alternative approach is provided by the structural kinetics models, amongst which the indirect structural kinetics model, which involves the use of a scalar structural parameter to describe the microstructural evolution, is the best known (Dullaert and Mewis, 2006; de Souza Mendes, 2009). The scalar structure parameter *ζ* takes values between zero and one, which represents a microstructure that is either completely broken down or completely associated. The use of *ζ* reduces the necessity for elaborate descriptions of the physical mechanisms happening at the particle length scales. The temporal evolution of the scalar structural parameter is dependent on both structural build-up and breakdown. The structural build-up is dependent on interparticle forces and/or Brownian forces, whereas the structural breakdown is driven by shearing. A generic formulation describing the evolution of *ζ* is given as:

*k*and

_{f}*k*represent constants that regulate the structural build-up and breakdown, respectively. The constants

_{b}*a*,

*b*,

*c*and

*d*are obtained from experimental data used for fitting (Mewis and Wagner, 2009). The first and second terms on the right-hand side of equation (1) represent the structure build-up and breakdown, respectively.

Nguyen and Boger (1985) used a rudimentary form of equation (1) to describe the thixotropic behaviour of bauxite residue suspensions. Other documented applications of the indirect structural theory include the description of bentonite mud (Billingham and Ferguson, 1993) and crude oils (Wachs *et al.*, 2009). Dullaert and Mewis (2006) considered a variant of equation (1) that is coupled with the evolution of elastic aggregate strain, while Toorman (1997) uses yield stress as an estimate of the scalar structure parameter. The structural theory has also been found to successfully explain the yield stress behaviour in measurements in gold mine tailings (Mizani and Simms, 2016). Grillet *et al.* (2009) and Mujumdar *et al.* (2002) used a variant of equation (1), where the structure build-up term independent of the shear rate is used to successfully capture the transient behaviour of viscosity observed in thixotropic fluids. To calculate the parameters associated with the structural model, curve-fitting can easily provide the parameter values owing to the simplicity of the model used by Nguyen and Boger (1985) and Toorman (1997). In the domain of surface mount technology (SMT), Mallik *et al.* (2010) used a simpler form of equation (1), which is integrated over time for a constant shear rate condition, to calculate the parameters associated with the structure model for various types of solder pastes. In Grillet *et al.*’s (2009) study, a non-linear curve-fitting method is used to estimate the parameter values owing to the complexity of the solution from the utilized model. The nonlinear curve-fitting techniques, like the Levenberg–Marquardt method, a variant of Gauss–Newton method, is sensitive to the initial guess used in the iterations (Transtrum and Sethna, 2012). The effect of the initial guess on the final estimate of the parameter is undesirable, especially for thixotropic fluids which has no previous published data.

Solder paste is an example of a thixotropic fluid which is widely used in the electronic industry for SMT applications. Understanding the rheological behaviour of solder paste is crucial in relating the printing performance of the paste with the flow and viscosity behaviour. In stencil printing, the solder paste undergoes shearing based on the pressure distribution produced by the squeegee (Krammer, 2015), and flow into a stencil aperture is driven by the resulting velocity distribution in the paste. The other parameters that are critical in the stencil printing process have been discussed by Durairaj *et al.* (2001). In jet printing, a piezoelectric stack drives the piston that ejects the solder paste onto the printed circuit board (Gu *et al.*, 2016). During the actuation process, which lasts less than a millisecond, the solder paste in the printer head experiences a wide range of shear stresses which causes the paste at different parts of the printer head to undergo a localized change in viscosity. The variations in the applied shear stress cause temporal changes of microstructure and viscosity during both stencil and jet printing of the solder paste. In other words, the flow of the solder paste, or other functional fluids, is intermittent, and as such, the volume flux will depend on the time-dependent rheological behaviour of the fluid. A number of studies have used shear thinning models like the Carreau–Yasuda model and cross-viscosity model to portray the thixotropic behaviour of these pastes (Durairaj *et al.*, 2002, Svensson *et al.*, 2016, Vachaparambil, 2016). Use of these simple viscosity models for numerical simulations of jet printing has shown substantial deviations from experimental observations, especially when comparing droplet morphology (Svensson *et al.*, 2016, Vachaparambil, 2016). In Mark *et al.*’s (2013) study, the Carreau–Yasuda model is used to describe the shear thinning behaviour of the solder paste along with a two-fluid model for assessing the granular suspension. The results showed good agreement with experimentally observed bulk velocities and jamming in a microfluidic contraction.

In this paper, a general method that can be used to estimate values of parameters of the indirect structure theory for any macroscopic suspension is proposed. The initial estimate of the parameters is obtained analytically and then fine-tuned by any nonlinear parameter fitting or optimization techniques; the latter approach is adopted in this paper. This work aims to provide a method to describe the rheological behaviour of any complex fluids and enable its characterization. The method is validated using experimental data for solder paste, owing to its widespread use and lack of rheological models to capture its complete thixotropic behaviour. The results show an improved prediction of thixotropic behaviour of the solder paste compared to the predictions based on the conventionally used shear thinning viscosity descriptions, such as the Carreau–Yasuda model. The main application of using a viscosity model that captures the complex viscosity behaviour, especially for the solder paste, is to aid in characterizing the fluid with respect to structure build-up and breakdown rates, for various industrial applications and numerical modelling.

## The analytical method to estimate the parameter values

The microstructural evolution defined using equation (2) is based on the generic formulation of evolution of *ζ* discussed earlier [equation (1)]:

The powers on the structural term *ζ* and (1 − *ζ*) are set to 1 to simplify the model. Similar approximations can be observed in Grillet *et al.* (2009), where the structure build-up term is dependent on shearing through the constant *k _{f}* and the breakdown term is modelled as in equation (2). The model in equation (2) provides a comprehensive definition of the transient nature of

*ζ*by including a structural build-up and breakdown term that has a nonlinear dependence on the shear rate. The model does not consider the effect of the Brownian motion, as the microstructural evolution in a macroscopic suspension can be assumed to be affected by only shearing. In reality, flocs or other structures can undergo elastic deformation owing to shearing, which results in a limit on the shear rate beyond which structural breakdown occurs (Mujumdar

*et al.*, 2002), but these effects are neglected in this paper.

Rearranging the terms in equation (2), for the ease of integration, results in:

Owing to the explicit dependence of structure build-up and breakdown terms on shear rate, both *k _{f}* and

*k*can be interpreted as constants and are therefore independent of γ̇. The parameters

_{b}*a*and

*b*are also assumed to be constants, making equation (3) a simple differential equation in

*ζ*whose solution, when integrated in time from 0 to

*t*under a constant shear rate and

*ζ*correspondingly ranging from

*ζ*

_{t}_{=0}to

*ζ*, can be written as:

The structure parameter can be related to viscosity as:

*η*

_{0}is the zero shear viscosity,

*η*is the infinite shear viscosity,

_{∞}*η*is the viscosity measured at a given

*ζ*and

*p*describes the non-linear relationship between viscosity and structure parameter (Grillet

*et al.*, 2009). If equations (4) and (5) are combined, the transient behaviour of viscosity under a constant shear rate γ̇ is described as:

In the limit *t* → ∞, the exponential term disappears from equation (6) and produces an equation for viscosity at steady state such that:

The expression in equation (7) resembles the Carreau–Yasuda model:

*λ*is a time constant and

*n*is the exponent. Comparing the Carreau–Yasuda model and equation (7), the following relations are observed:

The parameter *p* is also assumed to be independent of the applied shear as seen in equation (11). When *t* → ∞, the variation of *ζ* with time under constant shear rates as expressed in equation (4) can be written as:

Substituting equation (12) in equation (4) produces an expression for the structural parameter evolution under constant shear rate:

Combining equations (13) and (5) results in:

We can rewrite equation (14) as:

And substitute *k _{b}* in terms of

*k*using equation (9), such that:

_{f}Viscosity measurements at different shear rates at steady state provide data points for η_{1}(γ̇) and η_{2}(γ̇), respectively. If compared at the same time *t*, equation (16) can be written for two sets of data as described below:

The parameters *b* – *a* and *λ* are known from the Carreau–Yasuda model, and *a* can be calculated from equation (18):

Once *a* is known, the parameters *b*, *k _{f}* and

*k*can then be determined from equations (10), (16) and (9), respectively.

_{b}## Thixotropic fluid

A solder paste is a dense suspension consisting of a resin (carrier fluid) and metal alloy spheres. The paste is thixotropic by nature and is used in printed circuit board (PCB) assemblies (Puttlitz and Stalter, 2004). The main reason solder paste is used to validate the model is because the viscosity behaviour of the paste is critical in determining its application in the SMT and existing models to mathematically describe its rheology is insufficient. The metal alloy composition by mass seen in a standard lead-free alloy used in solder spheres consists of 3 per cent silver, 0.5 per cent copper and the remainder of tin (Figure 1). The size of the solder spheres ranges between 15 and 25 μm, which means the Brownian motion can be neglected for any operation at room temperature based on the calculation of the Péclet number. The solder paste, which belongs to the Type 5 category based on the solder sphere size, has a relative density of the spheres with respect to the flux equal to 8. The paste produced by Senju Metal Industry Co. Ltd., Japan, is 89 per cent solder spheres by weight.

## Rheological measurements

The most commonly used method to capture the rheological behaviour is through a shear/hysteresis loop test, which entails viscosity measurements where the shear rate increases from a minimum to a maximum value and then returns back to the minimum. The hysteresis loop test has been used to measure the thixotropic behaviour of the solder paste in Kolli *et al.*’s (1997) and Durairaj *et al.*’s (2004) studies. This measurement does not provide sufficient time for the viscosity to reach a steady value. Therefore, experimental data from a hysteresis loop cannot be used to determine the values of the constants of the analytical method.

An alternative method is to use steady shear rate measurements, i.e. a constant shear rate is applied until viscosity reaches a steady state. The steady shear rate experiments are performed in a plate–plate-type rheometer at 25°C, with the gap between the plates set at *d* = 0.2 mm. The distance between the plates *d* is based on a rule of thumb, such that the parallel plates must be separated by a gap of around or at least ten times the diameter of the solder spheres. The solder paste is given a total resting period of 15 minutes to allow for restructuring of the paste after it is injected onto the lower plate of the rheometer and to stabilize the forces from the plate surfaces. The data is collected for shear rates between 0.1 s^{–1} and 300 s^{–1}. Beyond this shear rate, the sample was lost owing to the high peripheral speed of the plate.

By curve fitting the data from multiple steady shear rate measurements using a Carreau–Yasuda model, the parameters *η*_{0}, *η _{∞}*,

*λ*and

*n*are determined (Table I). In Figure 2, the viscosity predicted by the Carreau–Yasuda model is plotted together with the experimental data obtained from the steady shear rate tests.

For the calculations, consider the parameters of the Carreau–Yasuda model (Table I) and the two points from the rheological data: γ̇_{1} = 0.1 s^{–1} and γ̇_{2} = 5 s^{–1}, which produced *η*_{1} = 4,135 Pa.s and *η*_{2} = 161 Pa.s, respectively. Following the steps mentioned for the analytical method, the parameters *a*, *b*, *k _{f}*,

*k*are calculated and tabulated in Table II.

_{b}## Results and discussion

The thixotropic behaviour predicted by the initial estimate of the parameters could be potentially different from the observed behaviour owing to the assumptions used. Therefore, the rheological behaviour of the solder paste is measured using a three interval thixotropy test (3IT tests). The 3IT tests has been successfully used in the understanding thixotropic behaviours of complex fluids (Dimic-Misic *et al.*, 2016; Toker *et al.*, 2015). The 3IT tests consists of three different shear conditions:

a reference interval, which consists of a low shear rate region;

a high shear rate region where structure is broken down; and

a regeneration interval, where the lower shear rate is reapplied to allow the rebuilding of the structure (Toker

*et al.*, 2015).

The 3IT tests used in this paper use an initial period of 16 s when the shear rate imposed is 1 s^{–1}, followed by a shear rate of 100 s^{–1} for 98 s and finally a shear rate of 1 s^{–1} for 94 s. In Figure 3, the measured viscosity is shown exhibiting a dependence on flow history, which is not predicted by the Carreau–Yasuda model. The parameters of the Carreau–Yasuda model used for comparison, in Figure 3, are based on the rheological model used by Svensson *et al.*(2016).

As measurements of viscosity at zero shear rate cannot be performed on a plate–plate rheometer, the initial shear rate is chosen to be 1 s^{–1} and the sample is given time to reach a steady viscosity value at the steady shear rate, which affects the initial structural parameter *ζ _{t}*

_{=0}. To consider the effects of starting the shearing from 1 s

^{–1}, the structural parameter for the first period, i.e. during the first 16 s, is calculated explicitly based on the viscosity measurements [equation (5)]. Once the shear rate changes, the structural evolution equation [equation (2)] is discretized as:

The discretization is performed such that the structure parameter at any given time *ζ _{n}* is dependent on the shear rate imposed at the given moment γ̇

_{n}and the flow history, which is taken into account through

*ζ*

_{n}_{−1}. The Δ

*t*used in the equation is calculated as

*t*−

_{n}*t*

_{n}_{−1}based on the time measurements from the experiments.

The rheological behaviour predicted by the structural model with the initial estimate of the parameters (Table II) is shown in Figure 4. The predicted viscosity behaviour exhibits a slower structural breakdown compared to the experimental data. The lower structural breakdown affects the viscosity when the structure rebuilds, and when the shear rate is removed or reduced. The difference between the experimental and predicted rheological behaviour is expected because of the assumptions used in obtaining the constitutive relationships for the parameters.

The initial estimate can be used as a starting point to obtain the optimized parameter values. An optimization based on the mean error between experimental and predicted viscosity behaviour is used. The mean error is used as the cost function that is to be minimized. Equations (9) and (10) show the relation between parameters *k _{f}*,

*k*,

_{b}*a*and

*b*when viscosity and shear rate have reached the steady state. These relationships would hold true even when the shear rate is time-dependent. The ratio of

*k*and

_{b}*k*is equal to

_{f}*λ*

^{b}^{-}

*, implying that whatever the value of*

^{a}*k*is used,

_{f}*k*would be adjusted, based on the value of

_{b}*λ*, to ensure that the relation is always maintained. Similarly, equation (10), which can be rewritten as

*b*= 2 +

*a*, shows how the value associated with

*b*is adjusted based on the assumption of

*a*. So calculating the rheological behaviour for a range of

*λ*and

*a*would reflect all permissible combinations of the parameters. Therefore, based on the initial estimates obtained from the analytical method (Table II), the value of

*k*is set equal to 2.4711 × 10

_{f}^{−8}, and the range of

*λ*and

*a*is set such that the starting values are slightly below the estimate the initial estimate obtained (Tables I and II), i.e.

*λ*is set to range between 600 and 1,500, and the parameter

*a*ranges from −15 to 0. The number of points in the range used for the parameters affects the accuracy of the mean error calculated. In this paper, an accuracy of 10

^{–3}on the mean error is used as a threshold to choose the number of points in the range. It is worth mentioning that beyond 730 data points in

*a*,

*b*,

*k*and

_{f}*k*, the results remains with the desired accuracy.

_{b}The rheological behaviour predicted by a combination of *λ* and *a* could lead to the structural breakdown rate being under-predicted in comparison to the experimental data. A constraint to neglect the values of *λ* and *a* for which the viscosity of the solder paste predicted by the structural theory, at the last time step is ±4 Pa.s more than the corresponding experimental data, is used. Further, a constraint is applied to ensure that the relation 0 ≤ *ζ* ≤ 1 is always maintained [equation (19)]. The resultant variation of the cost function for different values of *λ* and *a* is shown in Figure 5. And the cost function is minimum when *λ*, *a* and the corresponding mean error are 859.2593 s, −1.1934 and 22.6710 Pa.s, respectively. The optimized estimate of the parameters based on the *λ* and *a* from the minimization of cost function is tabulated in Table III. Figure 6 shows that the optimized estimate of the parameters can capture the thixotropic behaviour of the solder paste. The structural breakdown observed in the predicted behaviour matches the behaviour observed in the experiments. The structural build-up rate is under predicted even with the optimized estimate of the parameters, but the viscosity does match the experimental viscosity after a given time. The first-order relation describing microstructure evolution [similar to equation (2)] and its tendency to under-predict viscosity during structure build-up has already been shown in the work by Grillet *et al.* (2009) on another thixotropic fluid (polydisperse alumina in polyether triamine).

## Conclusions

A method to estimate the parameters associated with the indirect structural theory for any macroscopic suspension that exhibits thixotropy has been successfully developed and validated using a commercial solder paste. The proposed model succeeds in capturing the dependence of viscosity on flow history, which is not possible with simple viscosity models like Carreau–Yasuda model, relatively easily by considering a single set of parameters. The initial estimate of the parameters is based on the assumptions of the applied shear rate being constant. Multiple steady shear rate measurements were performed using plate–plate-type rheometer for the dense solder paste. The experimental data are used to determine the initial guess of parameters from the analytical method. Using an optimization algorithm to reduce the mean error between the experimental and predicted viscosity behaviour, new optimized estimates of the parameters are obtained. The optimized parameter estimate captures the thixotropic behaviour of the solder paste observed in experiments to a certain level of complexity. The model manages to match the experimental structure breakdown, but under-predicts the structural build-up. But the predicted viscosity matches the measurements from experiments after the microstructure is stabilized. The underprediction of the viscosity during the structure build-up when using indirect structure-based viscosity model has already been reported in previous literature.

The ability of the proposed model to capture the transient behaviour of the viscosity is extremely important, especially for applications in SMT. The proposed model could be coupled with CFD models to produce a more realistic behaviour of the solder paste during stencil or jet printing instead of simple viscosity models that fail to predict the thixotropic behaviour or granular flow models that are complicated to model. The model can also be used to characterize the flow of different solder pastes or other non-Brownian suspensions based on structure build-up and breakdown rates. Owing to the universality of the derivation, the methodology developed in the paper can also be used to estimate the parameters in the structural theory for any macroscopic suspension or complex fluid that exhibits a thixotropic behaviour.

## Figures

Parameter list for Carreau–Yasuda model obtained through curve fitting

Variable name |
Symbol | Value | Unit |
---|---|---|---|

Zero shear rate viscosity |
η_{0} |
1.474 × 10^{5} |
Pa.s |

Infinite shear rate viscosity |
η_{∞} |
9 | Pa.s |

Time constant |
λ |
640.4 | s |

Power-law exponent |
n |
0.1402 |

Initial estimate of parameters for the indirect structure-based viscosity model

Variable name |
Value |
---|---|

a |
−1.8328 |

b |
0.1672 |

k_{f} |
2.4711 × 10^{−8} |

k_{b} |
0.0101 |

p |
0.4299 |

Optimized estimate of parameters for indirect structure-based viscosity model

Variable name |
Value |
---|---|

a |
−1.1934 |

b |
0.8066 |

k_{f} |
2.4711 × 10^{−8} |

k_{b} |
0.0182 |

p |
0.4299 |

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