I am currently in queue to join a Master’s program in Chemistry, researching in a group that focuses in fundamental electrochemistry. Working in this field feels to me like racing a Daytona 500¹: always daunting, always feeling inexperienced, always mysterious. And that’s one of the reasons my coworkers and classmates always asked:

“Why is that you like electrochemistry so much?”

I’ll be honest: I have no clue. Writing a post like this is a way of looking into my heart and pointing at a certain reason I want to stay. So, open heart surgeries aside,

1. Electrochemistry is cryptic — in a wide sense, just as thermodynamics is and just how all physical chemistry arrives at being so after several decades of theories. However, any endeavor involving electricity in chemistry is quite rare outside this specific branch, so it is very welcome to work with something half-wet, half-dry.

2. Electrochemistry is mysterious: what kind of observable science lets you conjecture about the fate of trillions of trillions of atoms confined to picometer-thick surfaces, and produce anything near a feasible response? Again, one could argue in favor of physical chemistry overall – without denying it, there is an allure (to me) when electricity is involved.

   1. You’re very familiar, from an early age, with what heating or pushing things feels like – but electricity? Unless you live with someone working in the field as a technician or engineer, electricity is its own untamed beast. It’s a force you cannot easily control, that travels through air and vacuum² and requires special equipment to be correctly managed and detected.

   2. Even among early-career scientists, most of the mathematical joust required to understand mechanics to begin with, lacks most of the special tools that explain electromagnetic phenomena. For example, the Dirac distribution and all the frequency-domain jargon, Maxwell’s laws and whatnot.

3. To me, even if it’s old, electrochemistry is in its infancy. We do not know why steel shows an inductive loop when you record its impedance spectrum. We do not understand why the constant phase element begged to appear almost everywhere in impedance responses.Wwe do not know why nickel and its older brothers, palladium and platinum, are extremely affine to hydrogen and thus almost unbeatable at reducing bonds with it. Us being fascinated with such questions, even after 200 plus years of toying with them

4. What’s special about electrochemistry, and not physical chemistry overall or electromagnetism in physics, is the apparent bridge I’ve caught between the atomic scale (0.1 to 10 nm) and mesoscale (up to μm) in this branch. What you see is the complex interplay of surface phenomena: diffusion, (electro)sorption, redox reactions, transient EM fields… and the like. When you watch it in the nanometer window, it all makes no sense as a puzzling soup of equations; when you throw caution at the wind and average their behavior over time and space, electrochemical reactions appear as natural as the blue sky, erratic behaviors coalesce into a coherent set of steps, and breaking every bond³ becomes possible.

4. I easily become possessed by academic challenges, which is why I’m eager to begin my graduate studies. This doesn’t rule out physical chemistry, again, but that’s why my concentration is in this branch. Someone who loves symphonies may understandably have a preference for a wind or percussive instrument. However, they will without a doubt understand that, for their favorite instrument to sound accurate to the composer’s emotion, it has to be surrounded by other beautiful instruments, completing its flavor without relinquishing anything.

I’d sum up that this is how I see electrochemistry: after seeing the wonders of physics and chemistry, and falling enamoured of the mysteries of electricity, what comes after? What’s the symphony that will sound the loudest, the most mysterious, the most cryptic but yet the sweetest? In a poetic way, I chose electrochemistry because I don’t understand it, and even hoping for the best I’m not sure that many of its puzzles will be solved for how long I am alive. In the same spirit which by I prefer certain music, electrochemistry is so authentic and unique that it allures me to keep researching. That’s why I think it’s so great.

What oscillation, exactly?

One of my first experiments with a potentiostat was calibrating two silver / silver chloride (Ag/AgCl) reference electrodes (specifically, the Metrohm 6.0726.100 and 6.0733.100 models; say, RPint2 and RPint3). Both electrodes were being cared for in the upstairs lab, and my advisor commissioned me and one of her research assistants to calibrate their electrode potential, which for a KCl(aq) 3.0 M electrolyte should have been 209 mV at room temperature (which was very close to 25 °C)¹.

The (actually stable) reference was a well-aged Ag/AgCl electrode from the lab, ceremoniously tagged as “REP1” in bold black marker. Its formal electrode potential, , was determined to be 208.0 mV versus NHE² by a senior researcher long gone. Moreover, we connected the system to a very robust PGSTAT302N potentiostat to determine the calibration samples’ open circuit potential (OCP), leaving little to no room for error as we even detangled the cell-to-potentiostat wires³.

What’s so enchanting about electrochemistry is that nothing is entirely under our control, ever. Indeed, when we recorded the two electrodes’ OCPs against REP1, a strange pattern emerged for the three whole hours.

Do you see the snake-like oscillations in Figure 1, every around 2000 seconds and with a magnitude lower than 0.1 mV? It is such an unimportant problem, that doesn’t determine the potential determined (it averages out!), but it bothers me so, so much. Where does it actually come from?

No, really. Where does it come from?

My first intuition as an inexperienced electrochemist was to ask my advisor. Her contextual response was to tell me contractors were working outside the building, and that “this messes with the ground connection”. Since I don’t have any experience in electrical issues, I took her word for it at the time and didn’t worry too much. However, I already decided to investigate the trend these voltages could produce.

The shape of the curves suggest the voltages oscillate around a central value, so I wanted to test for two symmetrical distributions (i.e. uniform and normal). The Q-Q plots and the PPCC or Filliben’s test⁴ for both assumptions can be seen in Figure 2, and I’ve also gathered some summary statistics⁵ in Table 1.

Code: Implementation of the Filliben approximation in a QQ plot for normal and uniform distributions.

# Filliben's uniform order statistics medians
filliben <- function(data) {
  n <- length(data)
  store <- c()
  store[n] <- 0.5^(1/n)
  store[1] <- 1 - store[n]
  for (i in 2:(n-1)) {
    store[i] <- (i - 0.3175) / (n + 0.365)
  }
  return(store)
}

# Generate shifted uniform distributions
gen_unif <- function(data) {
  mi <- min(data)
  ma <- max(data)
  f <- 
    function(p) {
      qunif(p, min = mi, max = ma)
    }
  return(f)
}

# Store potentials in mV
rpaint2 <- ocp2$`Potential / V`[-(1:1000)] * 1e3
rpaint3 <- ocp3$`Potential / V` * 1e3
# Generate uniform distribution windows
up2 <- gen_unif(rpaint2)
up3 <- gen_unif(rpaint3)

# Graphics generation
layout(matrix(
  c(1,2,3,
    4,5,6), 2, 3, byrow = F
))

# histograms
hist(rpaint2, freq = F,
     xlab = "",
     ylab = "density", col = "lightpink1", border = "firebrick",
     breaks = seq(-0.7, -0.6, 0.5e-2), main = "histogram for OCP of RPint2")
hist(rpaint3, freq = F,
     xlab = expression(E[OCP] ~ " / mV"),
     ylab = "density", col = "lightblue1", border = "royalblue3",
     breaks = seq(-0.35, -0.10, 1e-2), main = "histogram for OCP of RPint3")

# normality Q-Q plots
qqnorm(rpaint2, 
       xlab = "", main = "normal Q-Q plot for RPint2",
       ylab = expression(E[OCP] ~ " / mV"),
       pch = 1, cex = 0.75, col = "gray50")
qqline(rpaint2, col = "firebrick",
       lty = 1, lwd = 2)

qqnorm(rpaint3, main = "normal Q-Q plot for RPint2",
       xlab = "theoretical quantiles",
       ylab = expression(E[OCP] ~ " / mV"),
       pch = 1, cex = 0.75, col = "gray50")
qqline(rpaint3, col = "royalblue3",
       lty = 1, lwd = 2)

# uniformity plots
qqplot(x = up2(filliben(rpaint2)), y = rpaint2, 
       xlab = "", main = "uniform Q-Q plot for RPint2",
       ylab = expression(E[OCP] ~ " / mV"),
       pch = 1, cex = 0.75, col = "gray50")
qqline(rpaint2, col = "firebrick", distribution = up2,
       lty = 1, lwd = 2)

qqplot(x = up3(filliben(rpaint3)), y = rpaint3, 
       xlab = "theoretical quantiles", main = "uniform Q-Q plot for RPint3",
       ylab = expression(E[OCP] ~ " / mV"),
       pch = 1, cex = 0.75, col = "gray50")
qqline(rpaint3, col = "royalblue3", distribution = up3,
       lty = 1, lwd = 2)

Code: Implementation of Filliben’s probability plot correlation coefficient (PPCC), plus the median and first four central moments of each dataset.

library(knitr)

# adjusted Fisher-Pearson skewness, G[1]
gskw <- function(x) {sqrt(1 - 1 / length(x)) / (length(x) - 2) * sum((x - mean(x))^3) / sd(x)^3}
# excess kurtosis wrt normal distribution, g[2]
ekrt <- function(x) {sum((x - mean(x))^4) / (length(x)) / sd(x)^4 - 3}
# Filliben's probability plot correlation test, or PPCC
ppcc <- function(data, dist = qnorm) {
  uordered <- filliben(data)
  quantiles <- dist(uordered)
  r <- cor(quantiles, sort(data))
  return(r)
}

summaries <- data.frame(
  "mean" = c(mean(rpaint2), mean(rpaint3)),
  "median" = c(median(rpaint2), median(rpaint3)),
  "stdev" = c(sd(rpaint2), sd(rpaint3)),
  "skw" = c(gskw(rpaint2), gskw(rpaint3)),
  "krt" = c(ekrt(rpaint2), ekrt(rpaint3)),
  "rn" = c(ppcc(rpaint2), ppcc(rpaint3)),
  "ru" = c(ppcc(rpaint2, up2), ppcc(rpaint3, up3))
)
rownames(summaries) <- c("RPint2", "RPint3")

kable(summaries, digits = 4,
      col.names = c("mean  $\\bar x$", 
                    "median $p_{50}$", 
                    "st. dev. $s$", 
                    "skewness $G_1$", 
                    "kurtosis $g_2$",
                    "normal ppcc $r_{x\\mathcal{N}}$",
                    "uniform ppcc $r_{xU}$"))

	mean	median	st. dev.	skewness	kurtosis	normal ppcc	uniform ppcc
RPint2	-0.6403	-0.6439	0.0171	0.2788	-1.0209	0.9775	0.9927
RPint3	-0.2309	-0.2319	0.0187	0.1682	-0.8424	0.9875	0.9934

It’s clear that both voltages follow, if any, an uniform distribution around their formal potentials. Since the emergent electrical noise from a potentiostat should follow a normal distribution⁶, this gives grounds (pun not intended!) to point to somewhere else. And since I have no expertise in electrical engineering…

There’s a spectrum of possibilities

…I happened to land on a beautifully written technical report from the North American SynchroPhasor Initiative⁷, that described how to diagnose and analyze the transient responses of a power system that’s oscillating. To me, this is sufficient to understand that

1. this system is oscillating way too slowly for an overtone of the power network to be happening. Even when a power system is mechanically forced, the resulting noise should be way faster. So, the construction workers are not to blame, and certainly not the 220 VAC network in place.

2. anything oscillating so slowly should be in the ballpark of pink noise, i.e. a signal with a power spectral density decaying as , where is a constant and is the frequency⁸.

Being nowhere closer to being an electrical engineer, I decided to brush up on Fourier transforms and ask someone else⁹ how to analyze a visible trend in the power spectrum. After zero-filling and transforming the data, a pair of periodograms appear with what seems to be a borderline pink noise.

Regrettably, the sampling period is a very long , which leads to an inaccurate Nyquist frequency of 0.5 Hz. This only helps us decode extremely low frequency trends, but luckily, this is all we need.

Code: Fourier transform and cleanup of both datasets to perform a log-log linear regression, looking for a pink noise-like trend.

# zero-padding implementation
zero_fill <- function(x, times = 9) {
  long = times * length(x)
  x <- c(x, rep(0, times = long))
  return(x)
}

# zero filled time-domain data
zf_rpaint2 <- zero_fill(rpaint2, times = 19)
zf_rpaint3 <- zero_fill(rpaint3, times = 19)

# PSD estimations
paint2 <- spectrum(zf_rpaint2, log = "yes", plot = F)
paint3 <- spectrum(zf_rpaint3, log = "yes", plot = F)

# 1/f noise regression

bool2 <- (paint2$freq >= 1e-4) & (paint2$freq <= 1e-2)
bool3 <- (paint3$freq >= 1e-4) & (paint3$freq <= 1e-2)

pink2 <- lm(log10(paint2$spec[bool2]) ~ log10(paint2$freq[bool2]))
pink3 <- lm(log10(paint3$spec[bool3]) ~ log10(paint3$freq[bool3]))

layout(matrix(
  c(1,2), 1, 2, byrow = F
))

plot(paint2$spec ~ paint2$freq, log = "yx",
     type = "l", lwd = 1.5, col = "gray20",
     main = "power spectrum of RPint2",
     xlab = expression(frequency * "," ~ nu ~ (Hz)), ylab = expression("spectral density (" * V^2 * "/ Hz)")
     )
points(x = paint2$freq[bool2], y = 10^predict.lm(pink2),
       type = "l", lty = 1, lwd = 2, col = "royalblue3")

coef <- round(as.vector(pink2$coefficients), 4)
peq <- substitute(italic(S)(nu) == 10^int * nu^coef, list(int = coef[1], coef = coef[2]))
text(x = 1e-5, y = 0.5e-5, col = "gray0",
     peq, adj = 0)
rsq <- summary(pink2)$adj.r.squared
rex <- substitute(R^2 == rsq, list(rsq = round(rsq, 4)))
text(x = 1e-5, y = 3.3e-7, col = "royalblue3",
  rex, adj = 0)

plot(paint3$spec ~ paint3$freq, log = "yx",
     type = "l", lwd = 1.5, col = "gray20",
     main = "power spectrum of RPint3",
     xlab = expression(frequency * "," ~ nu ~ (Hz)), ylab = expression("spectral density (" * V^2 * "/ Hz)")
)
points(x = paint3$freq[bool3], y = 10^predict.lm(pink3),
       type = "l", lty = 1, lwd = 2, col = "firebrick")

coef <- round(as.vector(pink3$coefficients), 4)
peq <- substitute(italic(S)(nu) == 10^int * nu^coef, list(int = coef[1], coef = coef[2]))
text(x = 1e-5, y = 1e-9, col = "gray0",
     peq, adj = 0)
rsq <- summary(pink3)$adj.r.squared
rex <- substitute(R^2 == rsq, list(rsq = round(rsq, 4)))
text(x = 1e-5, y = 2e-11, col = "firebrick",
  rex, adj = 0)

What’s common between both is the bump between 10-100 μHz, which doesn’t align with an oscillation every 2000 s (i.e. 500 μHz). Moreover, the frequency exponents exceed 2, which suggests that what I’m seeing is not any more pink noise than it is my computer trying to show me a Dirac delta without exploding.