Measurement Methodsand Calculations

to Determine Internal Deposit Stress

Frank H. LeamanSpecialty Testing & Development,

Inc.York, PA

Methods for Deposit Stress Determination

Bent Strip (simple beam theory)

Spiral Contractometer

Simple Beam Theory

Where: N = Thickness of the plated coating (inches) T = Thickness of the test strip (inches) D = Deflection of the strip due to bending (inches) L = Length of the test section (inches) E = Modulus of elasticity of the test strip (lb/in2) I = Moment of inertia of the cross section of test strip about its neutral axis S = Stress in plated layer (lb/in2) 

Then: S = 4E (N+T) D 3NTL

Bent Strip Method (Initial Approach)

During the application of a coating, one end of the test piece is held in a fixed position and the other end is free to move.

It is difficult to measure the value for D.

Bent Strip Method (Different Approach)

A test piece split into two legs spreads outward due to the deposit stress

The deflection is easily read by placing the test piece over a scale

Calculate the deposit stress value by using a simple formula

Simple Beam Tensile and Compressive Stress

Tensile

Compressive

Compressive and Tensile Stress

Compressive Tensile

Stress Evaluation Using the Bent Strip Method

Test Strip in a Plating Cell

In-Site 1 Plating Cell

Ideal for small solution volumes and lab studies, particularly when working with precious metals

Bent Strip Test Piece Measuring Stand

Stress Evaluation Using the Bent Strip Method

Bent Strip Plating Test Cell

Test Strip Plating Cell with Accessories

Typical Deposit Stress Evaluation Plating Set-Up

Deposit Stress Calculations for Test Strips

The Stoney Formula: = E T² M δ 3 L² t

E = Modulus of elasticity of the substrate = 120,655 kg/cm².T = Thickness of the substrate in millimeters = 0.05077 mm. δ = 1/2 the distance between the test strip leg tips in mm. Example: 0.540 inch spread ÷ 2 x 25.385 mm/inch = 6.85 mm. = Stress in megapascals, MPa. Note: MPa x 145 = PSI.L = Length of substrate on which the deposit is applied in mm.For Deposit Stress Analyzer test strips, this value is 76.2 mm.t = Deposit average thickness in millimeters. M = Correction for modulus of elasticity difference between

the deposit and substrate: M = EDeposit ÷ ESubstrate = 206,900 ÷ 120,690 = 1.714= E (.05077 mm) ² M ( δ mm) = mm³= MPa

3(76.2 mm)²(.002538 mm) 44.21 mm³Deposit Stress in PSI = MPa x 145 = PSI

Note: MPa is Megapascals, kg/cm.²

Spiral Contractometer Existing Design

The test piece is a spiral. One end of the spiral is held, other end is free to move. As the free end moves, a dial registers the movement in degrees. The stress of the coating can be calculated.

Spiral on an Existing Contractometer

Spiral Plated on Existing Type Contractometer for Target Nickel Deposit Thickness of 500µ” in a Semi-bright Bath after 20 Seconds Wood’s Nickel Strike

Deposit Location Thickness, µ”

Outside Surface 410

Inside Surface 85

Deposit stress over a 2 minute strike =

26.4% less than the New DesignContractometer result

New Spiral Properties

New design spirals are constructed from 0.010 inch thick stainless steel and have a precise surface area of 13.57 in2.

Spirals mount on the contractometer in a way that the entire spiral plates from end to end and deposition of metal on the inside of spirals is minimal even if they are void of a masking material.

The average test deposit thickness is 500 microinches.

Properties and Plating Conditions for Spiral Contractometer Tests

Spiral Material Stainless Steel Spiral Surface Area, in2 13.57 Square Feet 0.0942 Amps per square foot 30 Amps 2.90 Stock Thickness, inches 0.010 Avg. Deposit Thickness, µ” 500 Plating Time, Minutes 21 Solution Temperature 140° ± 1° F

A new geometry solves problems related to an exposed interior that allows deposition of the applied deposit to occur on the inside surface. Interior deposits reverse the type of stress and reduce calculated results as much as 30%. Interior masking is critical.

The new design provides masking of the interior surface by geometry and enables spirals to be plated tip to tip so the plated surface area is a constant value.

Other advantages: Stainless steel inserts 30% glass filled nylon

construction which prevents thread damage and spiral slipping

More accurate results Saves time

Spiral Contractometer Equipment to Determine Internal Nickel Deposit Stress

Spiral Contractometer with calibration weights, support stand and spiral test pieces. Container 4” diameter and 10” height for nickel strike anode basket and bath (1750 ml)

Titanium Mesh Anode Basket 3.5” outside and 2.25” inside diameter, 8” high with support contact tabs and cover for Wood’s nickel strike

Titanium Mesh Anode Basket 5” outside and 4” inside diameter with support contact tabs and cover for the plating bath

Nickel anode buttons to fill the anode baskets Pyrex beaker 4000 ml for a nickel plating bath Support stand – designed to perfectly center over beaker Magnetic stirrer hot plate, 115 volt Digital temperature Controller pre-wired with probe to control ± 10 F Power Supply constant current, constant voltage, 0-5 amps, 0-30 volts Magnetic stirrer hot plate, 115 volt

Contractometer Stand, Anode Basket & Beaker

Contractometer Plating Set-Up

Data Recording for Spiral Contractometer Tests

Deposit weight in grams:________ ________

Kc degrees:________ ________

Kt degrees:________ ________

Degrees deflection caused by the deposit:________ ________

Spiral weight in grams:________ ________

Deposit weight in grams by subraction:________ ________

Deposit thickness in microinches:________ ________

Average Deposit Thickness Calculation in Inches

T = _________W____________ = Inches D (87.55 cm2) (2.54 cm/inch)

W = Grams of nickelD = Density of nickel = 8.90 g/cm3, andT = Deposit thickness in inches

For the new spirals plated on the new design contractometers, the constant spiral plated surface area is 13.57 in2 and the following shortened formula applies:

T = W = Inches 1979.2

Calculating Deposit Stress

Stress = 13.02 (D) (M) ÷ w x d = PSI

D = Degrees caused by the deposit,M = Modulus of Elasticity of the deposit ÷ that of the substrate = 206,897 ÷ 198,186 = 1.044 for nickel deposits over new spirals that are 0.010 inch thick,w = degrees Kt from spiral calibration if the stress is tensile or degrees Kc if the stress is compressive, andd = Deposit thickness in inches.

Calculation Example:S = 13.02 (26) (1.04897) ÷ 33 (0.000536) = 20,073 PSI

Modulus of Elasticity Values

Stock Material Cu-Fe Alloy Ni –Fe Alloy Ni-Fe Alloy Pure Ni ES* 120,690 144,830 179,310 206,900Stock Thickness, in 0.0020 0.0015 0.0010 0.0010

Metal ED** Values for M*** Cadmium 31,720 0.263 0.219 0.177 0.153 Chromium 248,280 2.06 1.71 1.39 1.20 Cobalt 206,897 1.72 1.43 1.15 1.00 Copper 117,240 0.971 0.810 0.654 0.567 Gold 74,480 0.617 0.514 0.415 0.360 Nickel 206,900 1.71 1.42 1.14 1.00 Platinum 146,900 1.22 1.02 0.819 0.710 Rhodium 289,650 2.40 2.00 1.62

1.400 Silver 75,860 0.629 0.524 0.423 0.367 Zinc 82,760 0.686 0.571 0.462 0.400   ES*, modulus of elasticity of substrate material in the Stoney Formula. ED**, Modulus of elasticity of deposit for use in modified Deposit Stress Analyzer and Stoney formulas. M***, modulus of elasticity of deposit ÷ modulus of elasticity of substrate for deposit stress determinations using the modified Deposit Stress Analyzer and Stoney Formulas.

A Frequent Mistake in Test Procedure

1 2 3Deposit Thickness

To Stock Ratio 1:20 1:20 1:5Stock Thickness, Inches0.010 0.002 0.002Deposit Thickness, µ Inches 500 500 100Minutes Plated 20 4 20Current Density, ASF 30 30 30Deposit Stress, PSI 14,060 14,127 6,865

Note: Extra thick deposits of the harder metals increases the degree of stiffness which results in lower proportional test strip spread.

Spiral Test Strips

Formulas for Bent Strip with One End Stationary*

Bent Strip Stress Curve For the comparison of equations that follow that apply to

calculating the internal deposit stress of applied metallic coatings over various substrate materials, the value of U = 8.5 units = 0.780 inch will consistently be used as a basis. It will be noted that the calculated internal deposit stress values vary from equation to equation, particularly where the equation fails to address Modulus of Elasticity differences between the substrate and the deposit.

Relationship between δ and Z. Example: For a given test strip, U = 8.5 units = 0.780 inch, and δ = U in inches x 25.385 mm/inch ÷ 2, so in this case δ = 9.90 mm.

δ = 4Z Z = δ ÷ 4 L = 76.155 mm

Using δ = 9.900 mm, Z = 9.90 mm ÷ 4 = 2.475 mm

R = L² + 4Z² = 5824.1 = 303.34 mm 8Z 19.2*Note: These formulas only work for bent strip

applications and are not applicable for the spiral contractometer method.

Stoney Formula Without and With Correction for Modulus of Elasticity Differences Between the Deposit and the Substrate Example: For a Cu-Fe test strip, U = 8.5 units = 0.780 inch

δ = U in inches x 25.385 mm/ inch ÷ 2 = 9.900 mm

WITHOUTσ = 4ET²Z = ET² δ = 91.137 MPa = 13,214.9 PSI

3L²t 3L²tL = test strip plating length = 76.2mm,T = Stock thickness = 0.05077mm andt = Deposit thickness = 0.000075 inch = 0.001904mm

WITH M = Edeposit ÷ Esubstrate = 206900 ÷ 120690 = 1.714σ = ET² δM = 120690(0.05077)²(9.900mm)(1.715) = 156.30 MPa 3 L² 3(76.2mm)²(0.00194mm)

σ = MPa (145 PSI/MPa) σ = 22,663.5 PSI 

Other Bent Strip Formulas to Determine Internal Deposit Stress in Applied Metallic Coatings

Barklie and Davies Formulaσ = ET²

6Rt (1 – t/T) Heussner, Balden and Morse Formula

σ = 4ET²Z 3t (T + t) L

Brenner and Senderoff Formulasσ = ET(T+ ᵦt) ᵦ = Edeposit ÷ Esubstrate

6Rtσ = E (t + T)³

3Rt (2T + t)

Brenner and Senderoff Formula for Bent Strip Applications

Brenner and Senderoff Formula

σ = ET(T+ ᵦt) ᵦ = Edeposit ÷ Esubstrate = 1.714

6Rtσ = 120690 MPa (.05077mm)(.05077mm +1.714(.001904mm) = 95.538 MPa

6(303.34mm)(.001904mm)

95.538 MPa x 145PSI/MPa = 13,853 PSI

Note: This formula doesn’t correct for large differences in Modulus of Elasticity values. The uncorrected Stoney result was 13,215 PSI. To be correct, this Brenner and Senderoff formula requires modification as follows:

σ = ET²ᵦ = 22,310 PSI where ᵦ = 1.714

6Rt