A Method to Predict the Thickness of Poorly-Bonded Material Along Spray and Spray-Layer Boundaries in Cold Spray Deposition

Abstract

Multi-traverse CS provides a unique means for the production of thick coatings and bulk materials from powders. However, the material along spray and spray-layer boundaries is often poorly bonded as it is laid by the leading and trailing peripheries of the spray that carry powder particles with insufficient kinetic energy. For the same reason, the splats in the very first layer deposited on the substrate may not be bonded well either. A mathematical spray model was developed based on an axisymmetric Gaussian mass flow rate distribution and a stepped deposition yield to predict the thickness of such poorly-bonded layers in multi-traverse CS deposition. The predicted thickness of poorly-bonded layers in a multi-traverse Cu coating falls in the range of experimental values. The model also predicts that the material that contains poorly bonded splats could exceed 20% of the total volume of the coating.

Appendices

Appendix 1

Figure 12 schematically shows the measured thickness profile (solid curve), together with the corresponding ideal profile (broken curve). The value of η _l , the average deposition efficiency in spray periphery (i.e., x > x _o ), is calculated as η _l  = A/B where A is the area under the actual thickness and B is the area under the ideal Gaussian thickness profile given by Eq 16. By approximating the actual thickness curve at x > x _o by a straight line that hits the x-axis at x _o + R, A is calculated as

$$A = \frac{{R\tau (x_{o} )}}{2}$$

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B is calculated by integrating the ideal Gaussian thickness curve at x > x _o,

$$\begin{aligned} B & = \frac{{\mu_{m} }}{\rho \nu }\sqrt {\frac{\pi }{\alpha }} \int_{{x_{o} }}^{\infty } {e^{{ - \alpha x^{2} }} } {\text{d}}x \approx \frac{\tau (0)}{{\eta_{h} }}\sqrt {\frac{\pi }{\alpha }} \int_{{x_{o} }}^{\infty } {e^{{ - \alpha x^{2} }} } {\text{d}}x = \frac{{\tau (x_{o} )}}{{\eta_{h} \xi_{o} }}\sqrt {\frac{\pi }{\alpha }} \int_{{x_{o} }}^{\infty } {e^{{ - \alpha x^{2} }} } {\text{d}}x \\ & = \frac{{\tau (x_{o} )}}{{2\eta_{h} \xi_{o} }}\sqrt {\frac{\pi }{\alpha }} \left[ {1 - erf\sqrt { - \ln \xi_{o} } } \right] \\ \end{aligned}$$

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where τ(0) and (x _o ) are the values of actual thickness determined at x = 0 and x _o, respectively, and $\xi_{o} = e^{{ - \alpha x_{o}^{2} }}$. Thus,

$$\frac{{\eta_{l} }}{{\eta_{h} }} = \sqrt {\frac{\alpha }{\pi }} \frac{{R\xi_{o} }}{{1 - erf\sqrt { - \ln \xi_{o} } }}$$

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For the thickness profile in Fig. 1(b), R = 1.6 mm, ξ _o  = 0.315 and α = 0.1000, which yield η _l /η _h  = 0.70.

Fig. 12

Schematic cross-sectional profiles of ideal and actual thickness of a single-traverse coating. The areal ratio A/B equals the average deposition efficiency η _l at x > x _o

Appendix 2

When a stationary axisymmetric cold spray with a Gaussian mass flow distribution μ(r) = μ(x, y) = μ _m exp[ − α(x ² + y ²)] deposits mass on a flat substrate normal to the spray axis with a uniform deposition efficiency η, the deposited mass increases at a rate M′ given by

$$M^{\prime} = \eta \rho \iint {D\left( {x,y} \right)}{\text{d}}x{\text{d}}y$$

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Equation 22 is integrated by substitution, ξ = exp(− αr ²), to yield

$$M^{\prime} = \frac{{\eta \mu_{m} }}{\alpha }\int_{0}^{1} {\pi r^{2} } d\xi = - \frac{{\eta \pi \mu_{m} }}{\alpha }\int_{0}^{1} {\ln \xi d\xi } = \frac{{\eta \pi \mu_{m} }}{\alpha }$$

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If the spray has the duplex deposition rates illustrated in Fig. 7, M′ can be calculated by dividing the deposited mass into the three parts shown in Fig. 13: the dome A and the disk B in the core region (|r| ≤ r _o) where the deposition efficiency is high at η _h and the peripheral regions C (|r| > r _o) where the deposition efficiency is low at η _l . By applying Eq 23 to A and C,

$$\begin{aligned} M^{\prime} & = M^{\prime}_{A} + M^{\prime}_{B} + M^{\prime}_{C} \\ & = \frac{{\eta_{h} \mu_{m} }}{\alpha }\left( {\int_{{\xi_{0} }}^{1} {\pi r^{2} } d\xi + \xi_{o} \pi r_{o}^{2} } \right) + \frac{{\eta_{l} \mu_{m} }}{\alpha }\left( {\int_{0}^{{\xi_{0} }} {\pi r^{2} } d\xi - \xi_{o} \pi r_{o}^{2} } \right) \\ & = \frac{{\mu_{m} \pi }}{\alpha }\left[ {\eta_{h} - (\eta_{h} - \eta_{l} )} \right]\xi_{o} \\ \end{aligned}$$

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Fig. 13

Mass deposition rate M′ of a spray with duplex deposition rates can be calculated in three parts: dome A and bottom disk B in the core region (r ≤ r _o) and the peripheral regions C (r ≤ r _o)

Since the mass flow rate of the spray M is πμ _m /α which is obtained by substituting η = 1 in Eq 23, the overall deposition efficiency of the duplex spray, Ψ = M′/M, is given by

$$\varPsi = \eta_{h} - \left( {\eta_{h} - \eta_{l} } \right)\xi_{o}$$

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