Analysis of refractive index difference method

B. PEI Adsorption

2. Analysis of refractive index difference method

An example adsorption calculation is shown below, with the assumptions being that the [PEI] remaining in solution= 0.05 wt% and that the SA = 100 m².

1) Take the difference (hence the term ‘difference method’) between the original [PEI]

in solution (0.1 wt%) and the remaining [PEI] in solution (0.05 wt%). This is in Equation 22.

0.10⁡𝑤𝑡% − [𝑃𝐸𝐼] = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒

0.10⁡𝑤𝑡% − 0.05⁡𝑤𝑡% = 𝟎. 𝟎𝟓⁡𝒘𝒕% (22)

2) Convert this difference from wt% to mg (Equation 23). This is the adsorption in mg.

0.05⁡𝑤𝑡% = 0.02⁡𝑔⁡𝑃𝐸𝐼

40⁡𝑔⁡𝑤𝑎𝑡𝑒𝑟∗ 100 ( 1

100) (40⁡𝑔) (1000⁡𝑚𝑔

1⁡𝑔 ) = 𝟐𝟎⁡𝒎𝒈 (23)

3) Divide the adsorption in mg by the total surface area (SA) to obtain the adsorption on a surface area basis in mg/m² (Equation 24).

𝑃𝐸𝐼⁡𝐴𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛⁡ (𝑚𝑔

𝑚²) = ⁡ 20⁡𝑚𝑔

100⁡𝑚² = 𝟎. 𝟐⁡𝒎𝒈

𝒎^𝟐 (24)

Thus, if there is 0.05 wt% PEI remaining in solution and the total surface area of powder is 100 m², the amount of PEI adsorbed is calculated to be 0.2 mg/m².

measurements for both concentration-centrifuge time suspensions (20 total measurements) were averaged and plotted with error bars as refractive index as a function of centrifuge time. S3 SiC and B3 B4C were used in this experiment. They were not used in the PEI adsorption experiment, but it is assumed that the powders used in the PEI adsorption experiment will act the same or similarly to the powders used in this experiment. Because the 5, 10, 15 and 20 wt% SiC slopes were similar, they are simply labeled as “SiC” and averaged. From this graph, it can be concluded that the refractive index measurements of water and B4C are consistent and do not vary with time because the slopes are nearly zero.

The average B4C value was well above the refractive index of water even after centrifuging.

It also does not change with centrifuge time. It does, however, appear to increase linearly with increasing B4C concentration. This is shown in Figure 34, where the values from Figure 33 were averaged and plotted as a function of powder added (wt%).

The refractive index of SiC behaves differently than that of B4C. The slope is 1-2 orders of magnitude greater than that of B4C. The average refractive index seems to vary more. Figure 34 shows that the refractive index does not increase as linearly as B4C. The refractive index values of SiC are closer to water than the B4C refractive index values.

Centrifuge Time (min)

0 20 40 60 80 100 120 140

Refractive Index

1.33250 1.33300 1.33350 1.33400 1.33450 1.33500

20 wt% B₄C Slope = -3.49*10^-7

15 wt% B₄C Slope = -2.31*10^-7

10 wt% B₄C Slope = -7.53*10^-8

5 wt% B₄C Slope = -1.59*10^-7

SiC

Slope = -1.81*10^-6 ± 8.45*10^-7 Water

Slope = 0.00*10⁰ S₃ SiC (SSA = 1.13 m²/g) B₃ B₄C (SSA = 0.31 m²/g)

Figure 33. Refractive index as a function of centrifuge time.

Powder Added (wt%)

0 5 10 15 20 25

Refractive Index

1.33280 1.33300 1.33320 1.33340 1.33360 1.33380 1.33400 1.33420 1.33440 1.33460 1.33480

B₃ B₄C (SSA = 0.31 m²/g) S₃ SiC (SSA = 1.13 m²/g)

B₄C SiC SLOPE 8.34E-05 1.18E-05 INTERCEPT 1.33298 1.33303

RSQ 1.000 0.749

Figure 34. Refractive index as a function of powder added (wt%).

The 20 measurements that Figure 33 takes the average from is shown in Figure 35.

This graph shows the data for a 10-minute centrifuge time. The other four centrifuge times (5, 20, 40, 80 min) are in the appendix.

It was observed that, if 10 consecutive measurements (without replacing the sample) were taken of each suspension, the refractive index would increase as the number of consecutive tests increased. The time from one measurement to the next was approximately one minute. Thus, 10 consecutive measurements took 10 minutes. This increase is more rapid in SiC than B4C. Since the B4C measurements increase more linearly than SiC and have a smaller slope than SiC, B4C could be dissolving into the supernatant during mixing and cen5trifuging. Mineralogy data shows that B3 contains granite and B2O3

Consecutive Test Number

1 2 3 4 5 6 7 8 9 10

Refractive Index

1.33280 1.33300 1.33320 1.33340 1.33360 1.33380 1.33400 1.33420 1.33440 1.33460 1.33480

Water - Slope = 3.64*10^-7

SiC - Slope = 2.06*10^-5 ± 1.97*10^-5 5 wt% B₄C - Slope = 7.27*10^-7

10 wt% B₄C - Slope = 5.45*10^-7 15 wt% B₄C - Slope = 1.42*10^-6 20 wt% B₄C - Slope = 4.67*10^-6

Figure 35. Refractive index of SiC and B4C as a function of consecutive test number (10-minute centrifuge, S3 SSA = 1.13 m²/g, B3 SSA = 0.31 m²/g).

It is proposed that the rule of mixtures can be used to determine the volume of SiC and B4C left in suspension after centrifugation. The system of equations in Equations 25 and 26 can be solved for the volume fraction of B4C and SiC. The refractive index of SiC and B4C are taken to be 2.6473 and 3.2316, respectively.^55,61

𝑛_𝑚 = ⁡ 𝑓_𝑤,𝑣(1.33297) +⁡𝑓_𝐵,𝑣(3.23160)

1 = ⁡ 𝑓_𝑤,𝑣+ 𝑓_𝐵,𝑣⁡ (25) 𝑛_𝑚 = ⁡ 𝑓_𝑤,𝑣(1.33297) +⁡𝑓_𝑆,𝑣(2.64370)

1 = ⁡ 𝑓_𝑤,𝑣+ 𝑓_𝑆,𝑣⁡ (26) The results from the rules of mixture calculations are shown in Table V and Figure 36. If a 100 ml suspension is assumed, each of the calculated volume fractions in Table V can be taken without further calculation. The amount of B4C left in suspension increases linearly with an r² value of ~ 1.0. Since 5 wt% B4C has 0.023 vol% remaining in

suspension, 10, 15 and 20% are respectively two, three and four times 0.023 vol%. It is possible that the smallest B4C particles are dissolving into solution during the mixing process and cannot be centrifuged out of suspension.

Since the refractive index of SiC samples increases with cumulative tests, the rule of mixtures can be used to calculate how much more SiC is in suspension after 10 consecutive measurements. The results show that as a function of wt%, the refractive index increases linearly after one measurement (r² = 0.994).

The refractive index of a 0.1 wt% PEI solution is 1.33320. The volume fraction of SiC and B4C necessary to reach that refractive index value, in a solution of just water and powder, is 0.017 vol% and 0.012 vol%, respectively. If 100 ml suspensions are assumed and the densities for SiC and B4C are respectively taken to be 3.20 g/cm³ and 2.52 g/cm³, the mass of SiC and B4C needed to bring the refractive index of the suspension to 1.33320 is 0.054 g SiC (0.017 vol%) and 0.030 g B4C (0.012 vol%). It does not take a large amount powder to increase the refractive index of the suspension, even after the suspension has been centrifuged.

Based on the particle size distributions for these powders (Figures 14 and 15), it is estimated that the size of the particles left in suspension are < 0.1 µm.

Table V. Equations Used to Find the Fraction of Powder Left in Suspension After Centrifuging. CT = Consecutive Test.

Category nmixture fB,v

0.1 wt% PEI 1.33320 0.012%

5% B4C 1.33340 0.023%

10% B4C 1.33382 0.045%

15% B4C 1.33421 0.065%

20% B4C 1.33465 0.088%

Category nmixture fS,v

0.1 wt% PEI 1.33320 0.017%

5% (CT 1) 1.33302 0.003%

5% (CT 10) 1.33326 0.022%

10% (CT 1) 1.33303 0.005%

10% (CT 10) 1.33330 0.025%

15% (CT 1) 1.33305 0.006%

15% (CT 10) 1.33342 0.035%

20% (CT 1) 1.33307 0.007%

20% (CT 10) 1.33335 0.029%

Powder added (wt%)

0 5 10 15 20 25

Suspended particles after centrifuging (vol%)

0.00 0.02 0.04 0.06 0.08 0.10

S₃ SiC (test 1) B₃ B₄C

SiC B4C SLOPE 2.58E-06 4.36E-05 INTERCEPT 2.22E-05 7.90E-06

RSQ 0.994 1.000

Figure 36. Volume percent of powder in suspension after centrifuging as a function of powder added. The volume percent was calculated using the rule of mixtures.

The temperature of the suspensions is proposed to be a reason why the refractive index increases with consecutive measurement (the refractive index increases with time).

The refractometer only performs a measurement when the refractometer cell is at 20.00 ± 0.01 °C. The refractometer manual displays the refractive index of water as a function of temperature. The data is plotted in Figure 37. Refractive index decreases as temperature increases. The refractive index is rising with increasing consecutive measurement (when the sample is not replaced), yet the temperature is staying constant.

The average refractive index at each consecutive measurement is plotted in Figure 38. It is used to calculate the temperature of the suspensions using Figure 37. The temperature initially starts at 18.7 °C and decreases to 15.8 °C as the refractive index increases. Since the refractometer only measures at 20.00 ± 0.01 °C, it seems that the temperature is not affecting the refractive index.

Temperature (°C)

0 10 20 30 40 50 60 70 80

Refractive Index

1.32200 1.32400 1.32600 1.32800 1.33000 1.33200 1.33400 1.33600

n = (-1.27*10^-6)*T² - (3.99*10^-5)*T + 1.33424 r² = 1.000

T = (-4.12*10⁵)*n² + (1.09*10⁶)*n - (7.19*10⁵)

Figure 37. Refractive index of water as a function of temperature.⁵⁶

Refractive Index

1.33300 1.33310 1.33320 1.33330 1.33340

Temperature

14 15 16 17 18 19 20

1 2

3 4

5 6

7 8

9 10

T = (-4.12*10⁵)*n² + (1.09*10⁶)*n - (7.19*10⁵)

Figure 38. Temperature of each consecutive measurement. Temperatures are calculated from the average refractive index for each consecutive measurement.

Stokes’s Law can be used to calculate how long it will take a particle to settle in a centrifuge tube. Equation 27 shows Stokes’s Law solved for the time it takes for a particle to settle.

𝑡 = 18𝜂𝑥

𝑑²𝑔(𝜌_{𝑠𝑜𝑙𝑢𝑡𝑒}− 𝜌_{𝑠𝑜𝑙𝑣𝑒𝑛𝑡})⁡ (27) Where t = settling time, η is the viscosity of water (8.90*10^-3 g/cm*s), x is distance (8.5 cm is the distance from the 40 ml mark of a centrifuge tube to its bottom), d is the particle diameter (assuming that the smallest particle in suspension is 0.1 µm or 100 nm), g is the force due to gravity (981 cm/s²), ρsolute is the density of the powder going into the suspension (3.20 g/cm³ for SiC and 2.52 g/cm³ for B4C) and ρsolution is the density of the solvent (this is water for all experiments in this experiment, 1.00 g/cm³).

Figure 39 shows the results if time is plotted as a function of g-force experienced by the particles. The three lines show settling as a function of g-force assuming the diameter is 1.00 µm, 0.10 µm and 0.01 µm. The particles were subjected to 11,000 rpm for 10

minutes in the centrifuge for the adsorption experiments. The equation to convert rpm to relative gravitational force (rcf) is listed in Equation 28.

𝑟𝑐𝑓 = 1.12 ∗ 𝑟𝑎𝑑𝑖𝑢𝑠 ∗ (𝑟𝑝𝑚

1000)² (28)

The centrifuge had a radius of 12.3 cm. 11,000 rpm is equivalent to 16,669 rcf, or 16,669*g. As the g-force increases, the settling time decreases considerably. With 0.1 µm being the minimum particle brought out of suspension, it would take 6.31 minutes of centrifuging to bring all the particles out of suspension. For comparison, it would take 631 minutes (10.5 hours) to bring particles with a diameter of 0.01 µm out of suspension. It would take 0.0631 minutes (3.79 seconds) to bring particles with a diameter of 1 µm out of suspension. The d² term in Equation 27 shows that particle diameter has a large effect on settling time.

g-force (m/s²)

10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

Settling time (min)

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

D = 1.00 µm D = 0.10 µm D = 0.01 µm

6.31 minutes

16,669*g (11,000 rpm)

Based on the data presented in this study, the PEI adsorption data for SiC and B4C should show evidence that the difference method using refractive index does not work for polyelectrolyte adsorption in ceramic powders. Even with 11,000 rpm centrifuging, there will still be particles remaining in suspension. Therefore, the refractive index will be higher and will yield incorrect adsorption values. The six adsorption plots in Figure 40 (SiC) and Figure 41 (B4C) each demonstrate that the refractive index difference method is not suitable for measuring polyelectrolyte adsorption in ceramic powders.

Figure 41. PEI adsorption in B4C.

PEI remaining in solution for SiC is plotted as a function of surface area added in Figure 40-a and calculated using Equation 20. Although there is a trend showing decreasing PEI left in solution as surface area added increases, some of these values are negative. It is impossible to have more PEI remaining in solution after adsorption, as indicated by the areas labeled as impossible states. It is also impossible to have negative PEI remaining in solution. The data is not suitable for calculating adsorption values.

PEI adsorption as a function of SiC added is shown in Figure 40-b (calculated using Equation 21). The maximum adsorption line represents 40 mg PEI per designated surface area of powder. The amount of PEI in each suspension is initially 40 mg. There cannot be more adsorption than 40 mg. There also cannot be negative adsorption.

Figure 40-c shows a graph modeled after the adsorption plot in Cesarano’s work.⁴³ PEI adsorption is plotted as a function of PEI added on a dry weight basis (dwb). The 100%

adsorption line is calculated by determining how much PEI would be adsorbed by adding 1% PEI on a dwb of SiC. This is shown in Equation 29. This plot shows that more PEI is adsorbed than added and shows that there is negative adsorption.

( 1⁡𝑔⁡𝑃𝐸𝐼 100⁡𝑔⁡𝑆𝑖𝐶)

(7. 69⁡𝑚1⁡𝑔⁡𝑆𝑖𝐶 )²

⁄ =(1000⁡𝑚𝑔⁡𝑃𝐸𝐼 100⁡𝑔⁡𝑆𝑖𝐶 )

(7.69⁡𝑚1⁡𝑔⁡𝑆𝑖𝐶 )²

⁄ = 𝟏. 𝟑𝟎⁡𝒎𝒈

𝒎^𝟐 (29)

In Figure 41-a, not only does the amount of PEI remaining in solution increase as a function of surface area (calculated from Equation 20), it also shows there is more PEI remaining in solution than there was initially. This is a physically impossible circumstance and questions the reliability of this usage of refractive index.

Figure 41-b shows the PEI adsorption in B1 B4C as a function of surface area (calculated from Equation 21). Although all the adsorption values fall within the maximum adsorption line, the adsorption values eventually become negative. This cannot occur.

Figure 41-c shows the PEI adsorption in B1 B4C as a function of PEI added on a dwb of B1 B4C, respectively. The 100% adsorption line is calculated from Equation 30 by determining how much PEI would be adsorbed by adding 1% PEI on a dwb of B4C. The absorption values become negative, which is impossible.

( 1⁡𝑔⁡𝑃𝐸𝐼 100⁡𝑔⁡𝑆𝑖𝐶)

(16. 37⁡𝑚1⁡𝑔⁡𝑆𝑖𝐶 )²

⁄ = (1000⁡𝑚𝑔⁡𝑃𝐸𝐼

100⁡𝑔⁡𝑆𝑖𝐶 )

(16.37⁡𝑚1⁡𝑔⁡𝑆𝑖𝐶 )²

⁄ = 𝟎. 𝟔𝟏⁡𝒎𝒈

𝒎^𝟐 (30)