0}b_{m}\ell_{F}(t)^{m}a^{m}y^{a}$ $\displaystyle=\frac{y^{a}}{yt}\sum_{m\geq
0}b_{m}\ell_{F}(tx)^{m}=\frac{y^{a}}{yt}\mathscr{E}_{F}(\ell_{F}(tx))=\frac{tx}{yt}\cdot
y^{a}$ $\displaystyle=\frac{x}{y}y^{a}=\frac{x}{y}\mu(x,F\mathrm{log}(y)),$

as desired.

The proof of the iterated $F$-derivative is similar. Indeed, note that 9
implies that for any $j\geq 0$, we have:

(10)
$(y\partial_{y})^{m}\left(\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\right)=(y\partial_{y})^{m}y^{a-j}=(a-j)^{m}y^{a-j}=(a-j)^{m}\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}.$

Assume that 8 holds for $n$; then:

$\displaystyle\nabla_{F,y}^{n+1}\mu(x,F\mathrm{log}(y))$
$\displaystyle=\nabla_{F,y}\nabla_{F,y}^{n}\mu(x,F\mathrm{log}(y))$ (11)
$\displaystyle=x(x+_{\widetilde{F}}\langle{-1}\rangle_{F}(t))\cdots(x+_{\widetilde{F}}\langle{-n+1}\rangle_{F}(t))\nabla_{F,y}\left(\frac{\mu(x,F\mathrm{log}(y))}{y^{n}}\right).$

The derivative on the right-hand side can be calculated as follows:

$\displaystyle\nabla_{F,y}\left(\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\right)$
$\displaystyle=\frac{1}{yt}\sum_{m\geq
0}a_{m}\ell_{F}(t)^{m}(y\partial_{y})^{m}\left(\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\right)$
$\displaystyle=\frac{1}{yt}\sum_{m\geq
0}a_{m}\ell_{F}(t)^{m}\left(\frac{\ell_{F}(tx)}{\ell_{F}(t)}-j\right)^{m}\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}$
$\displaystyle=\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\frac{1}{yt}\sum_{m\geq
0}a_{m}(\ell_{F}(tx)-j\ell_{F}(t))^{m}$
$\displaystyle=\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\frac{1}{yt}\mathscr{E}_{F}(\ell_{F}(tx)-j\ell_{F}(t)).$

Since

$\ell_{F}(tx)-j\ell_{F}(t)=\ell_{F}(tx)+\ell_{F}([-j]_{F}(t))=\ell_{F}(tx+_{F}[-j]_{F}(t))),$

this becomes

$\displaystyle\nabla_{F,y}\left(\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\right)$
$\displaystyle=\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\frac{1}{yt}\mathscr{E}_{F}(\ell_{F}(tx+_{F}[-j]_{F}(t)))$
$\displaystyle=\frac{\mu(x,F\mathrm{log}(y))}{y^{j}}\frac{x+_{\widetilde{F}}\langle{-j}\rangle_{F}(t)}{y}.$

Plugging this into 11, we get that

$\nabla_{F,y}^{n+1}\mu(x,F\mathrm{log}(y))=x(x+_{\widetilde{F}}\langle{-1}\rangle_{F}(t))\cdots(x+_{\widetilde{F}}\langle{-n+1}\rangle_{F}(t))(x+_{\widetilde{F}}\langle{-n}\rangle_{F}(t))\frac{\mu(x,F\mathrm{log}(y))}{y^{n+1}},$

as desired. ∎

######  Corollary 4.5.7.

There is a dotted map (which is a homomorphism over $R[\\![t]\\!]$) filling in
the following diagram:

$\textstyle{\hat{\mathbf{G}}_{t}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbf{G}_{m}^{\sharp,F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
F\mathrm{log}(y)}$$\textstyle{(\mathbf{G}_{a})_{R[\\![t]\\!]}.}$

###### Proof.

In the notation of 4.5.2, we need to show that
$\beta_{n}(F\mathrm{log}(y))\in\mathscr{O}_{\mathbf{G}_{m}^{\sharp,F}}$ for
every $n\geq 0$. To prove this, let us work over
$(R\otimes\mathbf{Q})[\\![t]\\!]$, and expand $\mu(x,F\mathrm{log}(y))$ as a
power series in $\frac{(y-1)^{n}_{s}}{n!_{F}}$ using Lemma 4.3.14. Evaluating
8 in Proposition 4.5.6 at $y=1$, we obtain

$\nabla_{F,y}^{n}\mu(x,F\mathrm{log}(y))|_{y=1}=x(x+_{\widetilde{F}}\langle{-1}\rangle_{F}(t))\cdots(x+_{\widetilde{F}}\langle{-n+1}\rangle_{F}(t)).$

It follows from Lemma 4.3.14 that

$\displaystyle\sum_{n\geq 0}\beta_{n}(F\mathrm{log}(y))x^{n}$
$\displaystyle=\mu(x,F\mathrm{log}(y))$ (12) $\displaystyle=\sum_{n\geq
0}x(x+_{\widetilde{F}}\langle{-1}\rangle_{F}(t))\cdots(x+_{\widetilde{F}}\langle{-n+1}\rangle_{F}(t))\frac{(y-1)^{n}_{s}}{n!_{F}}.$

Taking the coefficient of $x^{n}$ on the right-hand side expresses
$\beta_{n}(F\mathrm{log}(y))$ as an $(R\otimes\mathbf{Q})[\\![t]\\!]$-linear
combination of the divided powers $\frac{(y-1)^{n}_{s}}{n!_{F}}$; but since no
rational denominators appear, this in fact expresses
$\beta_{n}(F\mathrm{log}(y))$ as an $R[\\![t]\\!]$-linear combination of the
divided powers $\frac{(y-1)^{n}_{s}}{n!_{F}}$, as desired. ∎

######  Example 4.5.8.

When $F$ is the multiplicative formal group law, 12 reduces to the following
identity:

$\displaystyle\sum_{n\geq
0}\frac{\mathrm{log}_{q}(y)(\mathrm{log}_{q}(y)-(q-1))\cdots(\mathrm{log}_{q}(y)-(n-1)(q-1))}{n!}x^{n}$
(13) $\displaystyle=\sum_{n\geq
0}q^{-\binom{j}{2}}x(x-[1]_{q})\cdots(x-[n-1]_{q})\frac{(y-1)(y-q)\cdots(y-q^{n-1})}{[n]_{q}!}.$

Indeed, we have

$x+_{\widetilde{F}}[-n]_{q}=x+\frac{q^{-n}-1}{q-1}+(q^{-n}-1)x=q^{-n}x+[-n]_{q}=q^{-n}(x-[n]_{q}).$

The identity 13 was communicated to the first author by Michael Kural, and was
motivation for the more general 12.

######  Corollary 4.5.9.

The group structure on $\mathbf{G}_{m}^{\sharp,F}$ is well defined.

###### Proof.

Suppose that $y_{1}$ and $y_{2}$ both admit $F$-divided powers
$\frac{(y-1)^{n}_{s}}{n!_{F}}$; we need to show that the same is true of the
product $y_{1}y_{2}$. Since
$F\mathrm{log}(y_{1}y_{2})=F\mathrm{log}(y_{1})+F\mathrm{log}(y_{2})$, one can
express $\beta_{n}(F\mathrm{log}(y_{1}y_{2}))$ in terms of
$\beta_{n}(F\mathrm{log}(y_{1}))$ and $\beta_{n}(F\mathrm{log}(y_{2}))$. Using
the identity 12 in the proof of Corollary 4.5.7 shows that $y_{1}y_{2}$ must
also admit $F$-divided powers, as desired. ∎

The main result of this section is the following, which recovers 7 when $F$ is
the additive formal group law. Using Remark 4.3.10, one can in fact refine
Theorem 4.5.10 to remain true when the base $R[\\![t]\\!]$ (or rather
$\hat{\mathbf{G}}=\operatorname{Spf}R[\\![t]\\!]$) is replaced by the
universal formal group $\hat{\mathbf{G}}^{\mathrm{univ}}$ over the moduli
stack of formal groups over $p$-nilpotent rings.

######  Theorem 4.5.10.

There is a Cartesian square of group schemes over $R[\\![t]\\!]$:

(14)
$\textstyle{\mathbf{G}_{m}^{\sharp,F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
F\mathrm{log}(y)}$$\scriptstyle{\mathrm{can}}$$\textstyle{\hat{\mathbf{G}}_{t}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle{p}\rangle^{\ast}}$$\textstyle{(\mathbf{G}_{m})_{R[\\![t]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
y^{p}}$$\textstyle{(\mathbf{G}_{m}^{(1)})_{R[\\![t]\\!]}.}$

The right-vertical map is Cartier dual to the homomorphism
$p\underline{\mathbf{Z}}\to\hat{\mathbf{G}}_{t}$ sending
$p\mapsto\langle{p}\rangle_{F}(t)$. In particular, there is an extension

$0\to(\mu_{p})_{R[\\![t]\\!]}\to\mathbf{G}_{m}^{\sharp,F}\xrightarrow{{F\mathrm{log}}}\hat{\mathbf{G}}_{t}^{\vee}\to
0.$

###### Proof.

To check that the diagram commutes, we need to check that there is an equality
of elements of $\mathscr{O}_{\mathbf{G}_{m}^{\sharp,F}}$:

$y^{p}=\langle{p}\rangle^{\ast}(F\mathrm{log}(y)).$

Since $R$ is $p$-completely flat over $\mathbf{Z}_{p}$, there is an injection
$R[\\![t]\\!]\subseteq(R\otimes\mathbf{Q})[\\![t]\\!]$; so it suffices to
check the desired identity in
$(R\otimes\mathbf{Q})[\\![t]\\!]\\{\beta_{n}(y)\\}_{n\geq 0}$. By the
discussion in 4.5.2, $\langle{p}\rangle^{\ast}$ can be expressed as

$\displaystyle\langle{p}\rangle^{\ast}(z)$
$\displaystyle=\exp(z\widetilde{\ell}_{F}(\langle{p}\rangle_{F}(t)))=\exp\left(z\frac{\ell_{F}(t\langle{p}\rangle_{F}(t))}{t}\right)$
(15)
$\displaystyle=\exp\left(z\frac{\ell_{F}([p]_{F}(t))}{t}\right)=\exp\left(p\frac{\ell_{F}(t)}{t}z\right).$

Note that this is also $\exp\left(\frac{\ell_{F}([p]_{F}(t))}{t}z\right)$. It
follows that

$\displaystyle\langle{p}\rangle^{\ast}(F\mathrm{log}(y))$
$\displaystyle=\exp\left(p\frac{\ell_{F}(t)}{t}F\mathrm{log}(y)\right)=\exp\left(p\frac{\ell_{F}(t)}{t}\frac{t}{\ell_{F}(t)}\mathrm{log}(y)\right)=\exp(p\mathrm{log}(y))=y^{p},$

as desired.

To check that the square is Cartesian, first note that the horizontal maps are
surjective. This is clear for the Frobenius on
$(\mathbf{G}_{m})_{R[\\![t]\\!]}$. For the map $F\mathrm{log}$, define
$F\exp(z):=\exp(\frac{\ell_{F}(t)}{t}z)$, so that $F\exp(z)=\sum_{n\geq
0}\beta_{n}(z)$. There is a homomorphism
$\hat{\mathbf{G}}_{t}^{\vee}\to(\mathbf{G}_{m})_{R[\\![t]\\!]}$ sending
$z\mapsto F\exp(z):=\exp(\frac{\ell_{F}(t)}{t}z)$, and
$z=F\mathrm{log}(F\exp(z))$. Using 12 with $y=F\exp(z)$, one sees that $F\exp$
lands in $\mathbf{G}_{m}^{\sharp,F}$, i.e., that
$\frac{(F\exp(z)-1)^{n}_{s}}{n!_{F}}$ is well-defined in
$\mathscr{O}_{\hat{\mathbf{G}}_{t}^{\vee}}$. This implies that $F\mathrm{log}$
is surjective.

It remains to show that the kernel of
$F\mathrm{log}:\mathbf{G}_{m}^{\sharp,F}\to\hat{\mathbf{G}}_{t}^{\vee}$ is
isomorphic to $(\mu_{p})_{R[\\![t]\\!]}$. Observe that $F\mathrm{log}(y)=0$
implies that $\mathrm{log}(y)=0$, which happens (by the Cartesian square 7) if
and only if $y^{p}=1$. Conversely, if $y^{p}=1$, then

$p\cdot F\mathrm{log}(y)=F\mathrm{log}(y^{p})=0,$

which implies that $F\mathrm{log}(y)=0$. ∎

######  Example 4.5.11.

It follows from Theorem 4.5.10 that $\mathbf{G}_{m}^{\sharp,F}$ is an
extension of $\hat{\mathbf{G}}_{t}^{\vee}$ by $(\mu_{p})_{R[\\![t]\\!]}$. In
the case of the multiplicative formal group law over $R=\mathbf{Z}_{p}$, this
was studied in [Dri21]. Namely, in [Dri21, Section 5.3.1], it is shown that
there is an extension $\widetilde{G}_{Q}$ of
$(\hat{\mathbf{G}}_{m,q-1})^{\vee}$ by
$(\mu_{p})_{\mathbf{Z}_{p}[\\![q-1]\\!]}$, given by the functor

$\widetilde{G}_{Q}:R\mapsto\\{(q,x,u)\in R^{\times}\times W(R)\times
R^{\times}\mid q-1\text{ is nilpotent},1+\Phi_{p}([q])x=[u^{p}]\\}.$

Here, $W(R)$ denotes the ring of $p$-typical Witt vectors of $R$. Drinfeld
shows that the group scheme $\widetilde{G}_{Q}$ is isomorphic over
$\mathbf{Z}_{p}[\\![q-1]\\!]$ to $\mathbf{G}_{m}^{\sharp,F}$ (as extensions of
$(\hat{\mathbf{G}}_{m,q-1})^{\vee}$ by
$(\mu_{p})_{\mathbf{Z}_{p}[\\![q-1]\\!]}$).

As shown in [Dri21, Appendix D] (see also [Dev23b, Remark C.3] and Example
4.5.3), the Cartier dual $(\hat{\mathbf{G}}_{m,q-1})^{\vee}$ can be identified
with

$(\hat{\mathbf{G}}_{m,q-1})^{\vee}=\operatorname{Spf}\mathbf{Z}_{p}[\\![q-1]\\!]\left[y,\frac{\prod_{j=0}^{n-1}(y-j(q-1))}{n!}\right]_{n\geq
0}.$

By 15, the homomorphism $\langle{p}\rangle^{\ast}$ corresponds to the
invertible element

$\langle{p}\rangle^{\ast}(z)=\exp\left(p\frac{\mathrm{log}(q)}{q-1}z\right)=q^{pz/(q-1)};$

this element plays an important role in [Dri21]. Note that this can
alternatively be written as

$\langle{p}\rangle^{\ast}(z)=\sum_{n\geq
0}\frac{\prod_{j=0}^{n-1}(z-j(q-1))}{n!}[p]_{q}^{n}=\sum_{n\geq
0}\frac{\prod_{j=0}^{n-1}(pz-j(q-1))}{n!}.$

In this case, Theorem 4.5.10 therefore specializes to give a Cartesian square
over $\mathbf{Z}_{p}[\\![q-1]\\!]$:

$\textstyle{\widetilde{G}_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{G}_{m}^{\sharp,F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto\mathrm{log}_{q}(y)}$$\scriptstyle{\mathrm{can}}$$\textstyle{\hat{\mathbf{G}}_{m,q-1}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z\mapsto
q^{pz/(q-1)}}$$\textstyle{(\mathbf{G}_{m})_{\mathbf{Z}_{p}[\\![q-1]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
y^{p}}$$\textstyle{(\mathbf{G}_{m}^{(1)})_{\mathbf{Z}_{p}[\\![q-1]\\!]}.}$

######  Remark 4.5.12.

In [Dri21, Section 5] (in particular, [Dri21, Remark 5.7.5]), it is shown that
$(\hat{\mathbf{G}}_{m,q-1})^{\vee}$ is isomorphic to the group scheme
$W_{\mathbf{Z}_{p}[\\![q-1]\\!]}[F-(1+[q]+\cdots+[q]^{p-1})]$, where
$W_{\mathbf{Z}_{p}[\\![q-1]\\!]}$ is the Witt scheme over
$\mathbf{Z}_{p}[\\![q-1]\\!]$ and $[q]\in W(\mathbf{Z}_{p}[\\![q-1]\\!])$ is
the Teichmüller lift of $q$. For a more general formal group
$\hat{\mathbf{G}}_{t}$, the methods of this section might give a Witt vector
model for the Cartier dual $\hat{\mathbf{G}}_{t}^{\vee}$, but we have not
explored this direction.

######  Example 4.5.13.

If $F$ is the hyperbolic formal group law, then
$\ell_{F}(t)=\frac{1}{2}\mathrm{log}\left(\frac{t+1}{1-t}\right)$. Setting
$t=q-1$, 15 says that

$\langle{p}\rangle^{\ast}(z)=\exp\left(p\frac{\mathrm{log}(\frac{q}{2-q})}{2(q-1)}z\right)=\left(\frac{q}{2-q}\right)^{pz/2(q-1)}.$

The denominator of $2$ appearing in the exponent implies that the cases $p>2$
and $p=2$ behave markedly differently. Since
$\frac{q}{2-q}=1+(q-1)\cdot\frac{2}{2-q}$, one can alternatively write

$\langle{p}\rangle^{\ast}(z)=\sum_{n\geq
0}\frac{\prod_{j=0}^{n-1}(pz-2j(q-1))}{n!}\left(\frac{1}{2-q}\right)^{n}=\sum_{n\geq
0}\frac{\prod_{j=0}^{n-1}(z-2j(q-1))}{n!2^{n}}\frac{[p]_{q}^{n}}{(2-q)^{pn}}.$

Write $\hat{\mathbf{G}}_{h,q-1}$ to denote the formal group over
$\mathbf{Z}_{p}[\\![q-1]\\!]$ corresponding to the rescaled hyperbolic formal
group law. In this case, Theorem 4.5.10 specializes to give a Cartesian square
over $\mathbf{Z}_{p}[\\![q-1]\\!]$:

$\textstyle{\mathbf{G}_{m}^{\sharp,F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
F\mathrm{log}(y)}$$\scriptstyle{\mathrm{can}}$$\textstyle{\hat{\mathbf{G}}_{h,q-1}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z\mapsto\left(\frac{q}{2-q}\right)^{pz/2(q-1)}}$$\textstyle{(\mathbf{G}_{m})_{\mathbf{Z}_{p}[\\![q-1]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
y^{p}}$$\textstyle{(\mathbf{G}_{m}^{(1)})_{\mathbf{Z}_{p}[\\![q-1]\\!]}.}$

######  Example 4.5.14.

Let $F$ be the formal group law over $\mathbf{Z}_{p}$ given by specializing
the formal group law of Example 4.3.12 to $v_{n}=1$. Let
$\hat{\mathbf{G}}=\operatorname{Spf}\mathbf{Z}_{p}[\\![t]\\!]$ denote the
associated formal group, and let $\hat{\mathbf{G}}_{t}$ denote the rescaled
formal group over $\mathbf{Z}_{p}[\\![t]\\!]$. Then, 15 says that
$\langle{p}\rangle^{\ast}(z)$ is a “$p^{n}$-typical” version of
$\exp\left(p\frac{\mathrm{Li}_{1/n}(t)}{t}z\right)$. If we write $E_{p}(t)$ to
denote the Artin-Hasse exponential, then $\langle{p}\rangle^{\ast}(z)$ can be
made explicit when $n=1$:

$\langle{p}\rangle^{\ast}(z)=E_{p}(t)^{pz/t}=\prod_{p\nmid
m}(1-t^{m})^{-\frac{p\mu(m)z}{tm}}.$

If we write $t={\widetilde{p}}$ (to keep with the notation of [BL22]), then
Theorem 4.5.10 in this case specializes to give a Cartesian square over
$\mathbf{Z}_{p}[\\![{\widetilde{p}}]\\!]$:

$\textstyle{\mathbf{G}_{m}^{\sharp,F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
F\mathrm{log}(y)}$$\scriptstyle{\mathrm{can}}$$\textstyle{\hat{\mathbf{G}}_{\widetilde{p}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z\mapsto
E_{p}({\widetilde{p}})^{pz/{\widetilde{p}}}}$$\textstyle{(\mathbf{G}_{m})_{\mathbf{Z}_{p}[\\![{\widetilde{p}}]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y\mapsto
y^{p}}$$\textstyle{(\mathbf{G}_{m}^{(1)})_{\mathbf{Z}_{p}[\\![{\widetilde{p}}]\\!]}.}$

This can be viewed as a $p$-typical version of Example 4.5.11.

######  Remark 4.5.15.

The canonical homomorphism
$\mathrm{can}:\mathbf{G}_{m}^{\sharp,F}\to(\mathbf{G}_{m})_{R[\\![t]\\!]}$ of
group schemes defines a quotient group stack
$\mathbf{G}_{m}^{F\mathrm{dR}}:=(\mathbf{G}_{m})_{R[\\![t]\\!]}/\mathbf{G}_{m}^{\sharp,F}$
over $R[\\![t]\\!]$. One can prove by direct calculation that the derived
global sections of the structure sheaf of the stack
$\mathbf{G}_{m}^{F\mathrm{dR}}$ calculates the $F$-de Rham complex
$F\Omega_{\square,\mathbf{G}_{m}}:=F\Omega_{\square,\mathbf{A}^{1}}\otimes_{R[\\![t]\\!][y]}R[\\![t]\\!][y^{\pm
1}]$. The main point (we omit the argument here) is that there is a quasi-
isomorphism of $R[\\![t]\\!]$-coalgebras

(16) $R[\\![t]\\!][y^{\pm
1}]\otimes_{F\Omega_{\square,\mathbf{G}_{m}}}R[\\![t]\\!][y^{\pm 1}]\simeq
R[\\![t]\\!]\left[y_{1}^{\pm 1},y_{2}^{\pm
1},\frac{(y_{1}-y_{2})^{n}_{s}}{n!_{F}}\right]_{n\geq
0}\simeq\mathscr{O}_{\mathbf{G}_{m}^{\sharp,F}\times_{\operatorname{Spf}R[\\![t]\\!]}\mathbf{G}_{m}},$

where the tensor product on the left-hand side is derived, and both sides are
implicitly $(p,t)$-adically completed. The final equivalence follows by noting
that $(y_{1}-y_{2})^{n}_{s}=y_{2}(y_{1}y_{2}^{-1}-1)^{n}_{s}$, so adjoining
$\frac{(y_{1}-y_{2})^{n}_{s}}{n!_{F}}$ is equivalent to adjoining the
$F$-divided powers $\frac{(y_{1}y_{2}^{-1}-1)^{n}_{s}}{n!_{F}}$.

In the case of the additive formal group law, 16 boils down to the
$p$-complete equivalence

$\mathbf{Z}_{p}[y^{\pm 1}]\otimes_{\mathrm{dR}_{\mathbf{Z}_{p}[y^{\pm
1}]/\mathbf{Z}_{p}}}\mathbf{Z}_{p}[y^{\pm
1}]\simeq\mathrm{dR}_{\mathbf{Z}_{p}[y^{\pm 1}]/\mathbf{Z}_{p}[y_{1}^{\pm
1},y_{2}^{\pm 1}]}\simeq\mathbf{Z}_{p}[y_{1}^{\pm 1},y_{2}^{\pm
1}]\langle{y_{1}-y_{2}}\rangle\cong\mathscr{O}_{\mathbf{G}_{m}^{\sharp}\times_{\operatorname{Spf}\mathbf{Z}_{p}}\mathbf{G}_{m}}$

which arises via [Bha12, Proposition 8.5]. In the case of the multiplicative
formal group law, 16 was shown in [Pri19].

Given this, Theorem 4.5.10 can be rephrased as the following two statements:

  1. (1)

The canonical map
$(\mathbf{G}_{m})_{R[\\![t]\\!]}\to\mathbf{G}_{m}^{F\mathrm{dR}}$ factors
through the Frobenius
$(\mathbf{G}_{m})_{R[\\![t]\\!]}\to(\mathbf{G}_{m}^{(1)})_{R[\\![t]\\!]}$.

  2. (2)

The map
$(\mathbf{G}_{m}^{(1)})_{R[\\![t]\\!]}\to\mathbf{G}_{m}^{F\mathrm{dR}}$ is
surjective, and its kernel is isomorphic to the Cartier dual of the rescaled
formal group $\hat{\mathbf{G}}_{t}$. Here, the map
$\hat{\mathbf{G}}_{t}^{\vee}\to(\mathbf{G}_{m}^{(1)})_{R[\\![t]\\!]}$ is
Cartier dual to the homomorphism
$p\underline{\mathbf{Z}}\to\hat{\mathbf{G}}_{t}$ sending
$p\mapsto\langle{p}\rangle_{F}(t)$. In particular, there is an isomorphism

$\mathbf{G}_{m}^{F\mathrm{dR}}\simeq(\mathbf{G}_{m}^{(1)})_{R[\\![t]\\!]}/\hat{\mathbf{G}}_{t}^{\vee}$

over $\operatorname{Spf}R[\\![t]\\!]$.

In the case of the multiplicative formal group law, this picture has been
discussed in great detail in [Dri21]. As we hope to explain in future work,
the above two statements are natural consequences of the homotopy-theoretic
perspective on the $F$-de Rham complex (see Remark 4.3.24). We also hope that
Theorem 4.5.10 might be useful in understanding 4.3.22, and, in particular, in
understanding a “stacky approach” to $F$-de Rham cohomology (following [BL22,
Dri18, Dri22, Bha22]). Indeed, the $F$-divided power scheme
$\mathbf{G}_{m}^{\sharp,F}$ is a priori rather difficult to access algebro-
geometrically, but Theorem 4.5.10 says that it can be understood concretely in
terms of $(\mathbf{G}_{m})_{R[\\![t]\\!]}$ and the Cartier dual of the
rescaled formal group $\hat{\mathbf{G}}_{t}$.

Let us end by noting that the following is an immediate consequence of Theorem
4.5.10:

######  Corollary 4.5.16.

Let $(\mathbf{G}_{m}^{\sharp,F})^{\vee}$ denote the Cartier dual of
$\mathbf{G}_{m}^{\sharp,F}$. Then, there is a pushout square over
$R[\\![t]\\!]$:

$\textstyle{\underline{p\mathbf{Z}}_{R[\\![t]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\mapsto\langle{p}\rangle_{F}(t)}$$\textstyle{\hat{\mathbf{G}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\mathbf{Z}}_{R[\\![t]\\!]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbf{G}_{m}^{\sharp,F})^{\vee}.}$

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#  Enhancement of non-contact friction between metal surfaces induced by the
electrical double layer

A.I. Volokitin∗ Samara State Technical University, 443100 Samara, Russia

###### Abstract

Casimir and electrostatic non-contact friction between two gold plates, and a
gold tip and a gold plate, are calculated taking into account the contribution
of the electrical double layer. It is shown that in an extreme-near field
($d<10$nm) the contribution from the electrical double layer leads to the
enhancement of non-contact friction by many orders of magnitude in comparison
to the result of the conventional theory without this contribution. Casimir
and electrostatic friction dominate for short and large separations,
respectively. The calculated electrostatic friction is in good agreement with
experimental data. The results obtained open the way to detect the Casimir
friction using Atomic Force Microscope.

PACS: 44.40.+a, 63.20.D-, 78.20.Ci

##  I Introduction

Figure 1:  Origin of non-contact friction Figure 2:  The electrical double
layer on metal surface (schematic)Kang2014ElectrochimActa

One of the important achievement in nanotribology is the discovery of non-
contact friction between bodies without direct contactStipe2001PRL ;
Kiesel2011NatMat ; Kiesel2013NatMat ; Volokitin2007RMP ; Kiesel2015Nanoscience
; Volokitin2017Book . Beside of the fundamental significance of this
phenomenon, it has practical application in biology and quantum computing for
ultra sensitive force registration due to the link between friction and force
fluctuations established by the fluctuation-dissipation theorem. Origin of
non-contact friction is shown in Fig.1: (a) Casimir friction. A fluctuating
electromagnetic field created by quantum and thermal fluctuations of the
current density in one body induces the current density in other body. The
interaction between the fluctuating and induced current densities is an origin
of the Casimir force. When the bodies are in relative motion the induced
current lag behind and pull the fluctuating current back producing the Casimir
friction. A theory of the Casimir friction for relative sliding of two plates
was developed in Refs.Volokitin1999JPCM ; VolokitinPRB2008  using the
fluctuational electrodynamics developed by Rytov at.al Rytov1953 ; Rytov1967 ;
Rytov1987 . The Casimir friction exists even at zero temperature due to
quantum fluctuations and is denoted as quantum frictionPendry1997JPCM . In
contrast to the Casimir forces and radiative heat transfer, which were
measured in many experiments, the detection of the Casimir friction is still a
challenging problem for experimentalists. However, the Casimir frictional drag
force between quantum wells and graphene sheets, and the current-voltage
dependence of nonsuspended graphene on the surface of the polar dielectric
SiO2, were accurately described using the theory of the Casimir
frictionVolokitin2001JPCM ; Volokitin2013EPL ; Volokitin2017Z_Natur ;
Volokitin2011PRL . In Ref.Volokitin2016JETPLett ; Volokitin2016PRB  it was
shown that the Casimir frictional drag force produced by current density in a
graphene sheet can be detected using an AFM. However, at present there is no
direct mechanical detection of the Casimir friction. (b) Interaction of a
charged tip with an image charge produces an electrostatic friction during
motion of the tip. Theory of the electrostatic friction was developed in
Refs.Volokitin2005PRL ; Volokitin2006PRB . Although the Casimir and
electrostatic friction are both of electromagnetic origin, the detailed
mechanism is not totally clear. In the conventional theory, where the
reflection amplitudes are determined by the Fresnel’s formulas, calculated
friction is many orders of magnitudes smaller than experimentally observed
valuesStipe2001PRL ; Kiesel2011NatMat ; Kiesel2013NatMat . However the
friction is enhanced by many orders of magnitudes by ionic
adsorbatesVolokitin2005PRL ; Volokitin2006PRB ; Volokitin2003PRL .(c) The van
der Waals and electrostatic interactions between fluctuating surface
displacements produce a time-dependent stresses acting on the surfaces that
excite the acoustic phonons and produce phonon heat transfer and friction for
moving bodies.

In this article the Casimir and electrostatic friction are calculated between
gold plates, and a gold tip and a gold plate taking into account of surface
charge density related with the electrical double layer. Metal consists of
positive ions occupying the positions of the crystal lattice and mobile
electrons. The electrons can not move away from the crystal because the
positive nucleus exerts an attractive force. Thus, the surface of a clean
metallic material can be considered as the superposition of two thin layers;
one positively charged below the surface of the solid and the other negatively
charged adjacent on the surface. This charge separation zone constitutes the
electrical double layer (Fig.2). Approximating the double layer by two
opposite charged planes, the plane charge density can be estimated from the
relation $4\pi\sigma_{s}ed_{0}=\Delta\varphi$ where for gold the separation
between planes $d_{0}\approx 0.1$nm and the potential step due to the double
layerVolokitin2021PRB  $\Delta\varphi\approx 4.3$eV, thus $\sigma_{s}\approx
0.38$Cm-2. It is shown that in an extreme near-field ($d<10$nm) the Casimir
and electrostatic friction are enhanced by many orders of the magnitudes in
comparison to the case when the electrical double layer is not taken into
account, and they can be measured using the present state of art equipment.
Giant enhancement of heat transfer between gold surfaces in an extreme near-
field due to the electrical double layer was studied in Ref.Volokitin2021PRB .

##  II Casimir friction

According to the theory of the Casimir frictionVolokitin1999JPCM ;
VolokitinPRB2008 ; Volokitin2007RMP ; Volokitin2017Book , for two plates
separated by a vacuum gap with thickness $d$ and sliding with relative
velocity $v$, for $d<\lambda_{T}=c\hbar/k_{B}T$ and non relativistic velocity
$v\ll c$, the frictional force between surfaces is dominated by the
contribution from evanescent electromagnetic waves and determined by

$f_{x}=\hbar\int_{0}^{\infty}\frac{d\omega}{2\pi}\int_{q>k}\frac{d^{2}qq_{x}}{(2\pi)^{2}}e^{-2k_{z}d}\left[\frac{4\mathrm{Im}R_{1p}(\omega,q)\mathrm{Im}R_{2p}(\omega^{\prime},q)}{\mid
1-e^{-2k_{z}d}R_{1p}(\omega,q)R_{2p}(\omega^{\prime},q)\mid^{2}}\left[n_{2}(\omega_{2}^{\prime})-n_{1}(\omega)\right]+(p\rightarrow
s)\right],$ (1)

where $n_{i}(\omega)=[e^{\hbar\omega/k_{B}T_{i}}-1]^{-1},$
$\omega^{\prime}=\omega-q_{x}v$, $R_{p}$ and $R_{s}$ are the reflection
amplitudes for $p-$ and $s-$ polarized electromagnetic waves, $k=\omega/c$,
$k_{z}=\sqrt{q^{2}-k^{2}}$, $q>\omega/c$ is the component of the wave vector
parallel to the surface. To linear order in the sliding velocity $f_{x}=\gamma
v$ where the friction coefficient at $T_{1}=T_{2}=T$ is determined by
Volokitin1999JPCM ; VolokitinPRB2008 ; Volokitin2007RMP ; Volokitin2017Book

$\gamma_{rad}=\frac{\hbar^{2}}{8\pi^{2}k_{B}T}\int_{0}^{\infty}\frac{d\omega}{\mathrm{sinh}^{2}(\hbar\omega/2k_{B}T)}\int_{0}^{\infty}dk_{z}k_{z}(k_{z}^{2}+k^{2})e^{-2k_{z}d}\left[\frac{\mathrm{Im}R_{1p}(\omega,q)\mathrm{Im}R_{2p}(\omega,q)}{\mid
1-e^{-2k_{z}d}R_{1p}(\omega,q)R_{2p}(\omega,q)\mid^{2}}+(p\rightarrow
s)\right].$ (2)

In the presence of the surface charge density, that can be due to a potential
difference or the electrical double layer, the reflection amplitude for the p
-polarized electromagnetic waves is determined by Volokitin2019JETPLett

$R_{p}=\frac{i\varepsilon k_{z}-k_{z}^{\prime}+4\pi
iq^{2}\alpha_{s}\varepsilon}{i\varepsilon k_{z}+k_{z}^{\prime}-4\pi
iq^{2}\alpha_{s}\varepsilon}$ (3)

where $k_{z}^{\prime}=\sqrt{\varepsilon k^{2}-q^{2}}$, $\varepsilon$ is the
dielectric function of the substrate, and $\alpha_{s}$ is the normal component
of the surface dipole susceptibility. Without taking into account the
contribution of the electric double layer, Eq.(3) reduces to the Fresnel’s
formula for the reflection amplitude for $p$-polarized electromagnetic waves

$R_{p}=\frac{i\varepsilon k_{z}-k_{z}^{\prime}}{i\varepsilon
k_{z}+k_{z}^{\prime}}.$ (4)

Due to the screening by electrons in surface layer the interaction of an
external electromagnetic field with lower of the double layer plane can be
neglected. In this case $\alpha_{s}=\sigma_{s}^{2}M$ where $M$ is the surface
mechanical susceptibility that determines the surface displacement under the
action of external mechanical stress: $u=M\sigma_{zz}^{ext}$. In the elastic
continuum model Persson2001JPCM

$M=\frac{i}{\rho
c_{t}^{2}}\left(\frac{\omega}{c_{t}}\right)^{2}\frac{p_{l}(q,\omega)}{S(q,\omega)},$
(5)

where

$S(q,\omega)=\left[\left(\frac{\omega}{c_{t}}\right)^{2}-2q^{2}\right]^{2}+4q^{2}p_{t}p_{l},$
$p_{t}=\left[\left(\frac{\omega}{c_{t}}\right)^{2}-q^{2}+i0\right]^{1/2},\,\,p_{l}=\left[\left(\frac{\omega}{c_{l}}\right)^{2}-q^{2}+i0\right]^{1/2},$

where $\rho$, $c_{l}$, and $c_{t}$ are the mass density of the medium, the
velocity of the longitudinal and transverse acoustic waves. A surface charge
density and dipole moment is also induced by the applied potential
differenceVolokitin2019JETPLett . Due to the screening of the parallel
component of the electric field the contribution of the electrical double
layer to the reflection amplitude for $s$-polarized electromagnetic waves can
be neglected and for these waves the reflection amplitude is determined by the
the Fresnel’s formula

$R_{s}=\frac{ik_{z}-k_{z}^{\prime}}{ik_{z}+k_{z}^{\prime}}.$ (6)

##  III Phononic friction

The electrostatic and van der Waals interactions between surfaces 1 and 2
results in stresses that act on surfaces 1 and 2. If $u_{1}$ and $u_{2}$
denote the surface displacements of solid 1 and 2 then
equationsVolokitin2019JETPLett ; Volokitin2020JPCM

$\sigma_{1}(\omega)=au_{1}(\omega)-bu_{2}(\omega^{\prime}),$ (7)
$\sigma_{2}^{\prime})=au_{2}(\omega^{\prime})-bu_{1}(\omega),$ (8)

where $\omega^{\prime}=\omega-q_{x}v$,

$a=\frac{H}{2\pi
d^{4}}+q\sigma_{s}^{2}\mathrm{coth}qd,\,b=\frac{H}{4\pi}\frac{q^{2}K_{2}(qd)}{d^{2}}+\frac{q\sigma_{s}^{2}}{\mathrm{sinh}qd},$

where the first and second terms are due to the van der Waals and
electrostatic interactions, respectively, $H$ is the Hamaker constant,
$K_{2}(x)$ is the modified Bessel function of the second kind and second
order. The surface displacements due to thermal and quantum fluctuations are
determined by Volokitin2019JETPLett ; Persson2011JPCM

$u_{1}(\omega)=u_{1}^{f}(\omega)+M_{1}(\omega)[au_{1}(\omega)-bu_{2}(\omega^{\prime})],$
(9)
$u_{2}(\omega^{\prime})=u_{2}^{f}(\omega^{\prime})+M_{2}(\omega^{\prime})[au_{2}(\omega^{\prime})-bu_{1}(\omega)],$
(10)

where according to the fluctuation-dissipation theorem, the spectral density
of fluctuations of the surface displacements is determined by
LandauStatisticalPhysics

$\langle|u_{i}^{f}|^{2}\rangle=\hbar\mathrm{Im}M_{i}(\omega,q)\coth\frac{\hbar\omega}{2k_{B}T_{i}}$
(11)

where $M_{i}$ is the mechanical susceptibility for surface $i$. The friction
force can be calculated from equationVolokitin2017Book ; Volokitin2002PRB

$f_{x}v=Q_{1}+Q_{2}$ (12)

where $Q_{1}$ and $Q_{2}$ are the heat generated in surfaces 1 and 2 which are
determined by the rate of the work of the mechanical stress that act on
surfaces 1 and 2 in the rest reference frame of surfaces 1 and 2

$Q_{1}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int\frac{d^{2}q}{(2\pi)^{2}}\omega\mathrm{Im}\langle
u_{1}\sigma_{1}\rangle,$ (13)
$Q_{2}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int\frac{d^{2}q}{(2\pi)^{2}}\omega^{\prime}\mathrm{Im}\langle
u_{2}\sigma_{2}\rangle$ (14)

From Eqs.(7-14) we get

$f_{x}=4\int_{0}^{\infty}\frac{d\omega}{2\pi}\int\frac{d^{2}q}{(2\pi)^{2}}q_{x}\frac{b^{2}\mathrm{Im}M_{1}(\omega)\mathrm{Im}M_{2}(\omega^{\prime})}{\mid(1-aM_{1}(\omega))(1-aM_{2}(\omega^{\prime})-b^{2}M_{1}(\omega)M_{2}(\omega^{\prime})\mid^{2}}\left[n_{2}(\omega^{\prime})-n_{1}(\omega)\right].$
(15)

At $T_{1}=T_{2}=T$ the phonon friction coefficient is given by

$\gamma_{ph}=\frac{\hbar^{2}}{8\pi^{2}k_{B}T}\int_{0}^{\infty}\frac{d\omega}{\mathrm{sinh}^{2}(\hbar\omega/2k_{B}T)}\int_{0}^{\infty}dqq^{3}\frac{b^{2}\mathrm{Im}M_{1}(\omega)\mathrm{Im}M_{2}(\omega)}{\mid(1-aM_{1}(\omega))(1-aM_{2}(\omega)-b^{2}M_{1}(\omega)M_{2}(\omega)\mid^{2}}.$
(16)

##  IV Numerical results

Fig.3 shows the dependence of friction coefficient for the Casimir friction
between two gold plates on the distance between them for different mechanisms
at $T=300$K. For gold $c_{l}=3240$ms-1, $c_{t}=1200$ms-1, $\rho=1.9280\times
10^{4}$kgm-3, $H=34.7\times 10^{-20}$J (see Ref.Hamaker2015 ) and dielectric
functionChapuis2008PRB

$\varepsilon=1-\frac{\omega_{p}^{2}}{\omega^{2}+i\omega\nu},$ (17)

where $\omega_{p}=1.71\times 10^{16}$s-1, $\nu=4.05\times 10^{13}$s-1 . In
Fig.3 the blue lines are for the Casimir friction associated with the
electrical double layer and $p$-polarized waves. The green and pink lines are
for the phonon friction associated with the van der Waals and the
electrostatic interaction between the fluctuating surface displacements at the
potential difference $\varphi=10$V. The red line is the theory result for the
Casimir friction without the contribution from the double layer. Full and
dashed lines are for the contributions from the bulk and surface (Rayleigh)
acoustic modes. The mechanical susceptibility has a pole at
$\omega=\omega_{s}=c_{s}q=\xi c_{t}q$, where $\omega_{s}$ and $c_{s}$ are the
frequency and the propagation velocity of the Rayleigh surface waves, where
for gold $\xi=0.94$. Near the pole at $\omega\approx\omega_{s}$ the mechanical
susceptibility can be written asVolokitin2020JPCM

$M=-\frac{c}{\rho c_{t}(\omega-c_{s}q+i\gamma)},$ (18)

where for gold $c=0.36$ and $\gamma$ is the damping constant for surface wave
for which can be used estimateVolokitin2020JPCM ; Pendry2016PRB
$\gamma=0.17\omega$. In the extreme near-field ($d<10$nm) the double layer
contribution leads to an enhancement of the Casimir friction by many orders of
the magnitude in comparison with the theory which does not include this
contribution.

Figure 3: The dependence of the Casimir friction coefficient between two gold
plates on the distance between them at $T=300$K. The blue lines is the
contributions from fluctuating dipole moment of surface electrical double
layer at the double layer potential step $\Delta\varphi=4.3$eV. The brown
lines is the contributions from the fluctuating dipole moment induced on
surfaces by the potential difference $\varphi=10$V. The pink line is the
contribution from van der Waals interaction between fluctuating surface
displacements. The Full and dashed lines are the contributions from the bulk
and surface (Rayleigh) phonon modes. The red line is the result of the
conversional theory of the Casimir friction without the contributions from the
fluctuations of the surface displacements.

Vibration of a tip charged by bias voltage $\varphi$ parallel to metallic
substrate produces an electrostatic friction. In the case when the tip is a
section of the cylindrical surface with the radius of curvature $R$ and with
the width $w$ the friction coefficient is the same as for a wire located at
$d_{1}=\sqrt{(d+R)^{2}-R^{2}}$ and with the charge per unit length
$Q=C\varphi$, where $C^{-1}=2\ln[(d+R+d_{1})/R]$, and is given by
Volokitin2005PRL ; Volokitin2006PRB ; Volokitin2017Book

$\Gamma_{c}=\lim_{\omega\to
0}2C^{2}\varphi^{2}w\int_{0}^{\infty}dqqe^{-2qd_{1}}\frac{\mathrm{Im}R_{p}(\omega,q)}{\omega}$
(19)

where $R_{p}$ is the reflection amplitude for $p$-polarized waves. Using in
Eq.(19) the Fresnel’s formula for the reflection amplitude gives the friction
coefficient

$\Gamma_{c}^{Fresnel}=\frac{w\nu\varphi^{2}}{16\omega_{p}^{2}d^{2}}.$ (20)

For gold with $w=7\cdot 10^{-6}$ m, $d=20$ nm and $\varphi=1$V, Eq.(20) gives
$\Gamma_{c}=1.69\cdot 10^{-20}$ kg/s that is eight orders of magnitude smaller
than the experimental value $3\cdot 10^{-12}$kg/s from Ref.Stipe2001PRL . In
the case when the surface double layer is taken into account, for complete
screening, when the interaction of the external electric field with the
subsurface charge distribution is neglected, and $1/\varepsilon\ll 4\pi
q\sigma_{s}^{2}M\ll 1$ the reflection amplitude (3) can be approximated by

$R_{p}^{DL}\approx 1+8\pi q\sigma_{s}^{2}M$ (21)

and the friction coefficient is determined by

$\Gamma_{c}=\lim_{\omega\to
0}16\pi\sigma_{s}^{2}C^{2}\varphi^{2}w\int_{0}^{\infty}dqq^{2}e^{-2qd_{1}}\frac{\mathrm{Im}M(\omega,q)}{\omega}.$
(22)

The contribution to friction from bulk acoustic waves, which is determined by
the integration for $q\leq\omega/c_{t}$, vanishes in the limit
$\omega\rightarrow 0$. Thus at low frequency the electrostatic friction
coefficient is determined by Rayleigh surface waves. Substitution of (18) in
(22) for $R\gg d$ gives

$\Gamma_{c}^{DL}=0.14\frac{\sigma_{s}^{2}\varphi^{2}wR^{1/2}}{d^{3/2}\rho
c_{s}^{2}c_{t}}.$ (23)

The friction coefficient has the same distance dependence as it was observed
experimentallyStipe2001PRL . At $R=1\mu$m and with the same parameters as
above $\Gamma_{c}^{DL}=1.9\cdot 10^{-12}$kgs-1 is in good agreement
experimental value $3\cdot 10^{-12}$kg/s from Ref.Stipe2001PRL .

For a spherical tip the friction coefficient Volokitin2005PRL ;
Volokitin2006PRB ; Volokitin2017Book

$\Gamma_{s}=\lim_{\omega\to
0}\frac{C^{2}\varphi^{2}}{2}\int_{0}^{\infty}dqq^{2}e^{-2qd_{1}}\frac{\mathrm{Im}R_{p}(\omega,q)}{\omega}$
(24)

where

$d_{1}=\pm\sqrt{3Rd/2+\sqrt{(3Rd/2)^{2}+Rd^{3}+d^{4}}},\,\,C=\frac{d_{1}^{2}-d^{2}}{2d}$
(25)

Substituting in (LABEL:C) the Fresnel’s formula for the reflection amplitude
and using Eq.(21) gives

$\Gamma_{s}^{Fresnel}=\frac{\sqrt{3R}\nu\varphi^{2}}{4\omega_{p}^{2}d^{3/2}}$
(26)

and

$\Gamma_{s}^{DL}=0.14\frac{R\sigma_{d}^{2}\varphi^{2}}{d\rho c_{t}c_{s}^{2}}$
(27)

Figure 4: The dependence of the Casimir (full lines) and electrostatic (dashed
lines) friction coefficient on the distance between a gold tip and a gold
plate at $T=300$K. The blue and green lines include the contributions from the
electrical double layer for a spherical tip with radius $R=1$$\mu$m and a
cylindrical tip with radius $R=1$$\mu$m and width $w=7$$\mu$m, respectively.
The brown and pink lines are the results of the conventional theory for a
cylindrical and spherical tip, respectively.

Fig. 4 shows the dependence of the friction coefficient associated with the
friction between a gold tip and a gold plate on the separation $d$. The
friction coefficient for the Casimir friction in the tip-plate configuration
can be obtained from the plate-plate configuration using the proximity (or
Derjyaguin) approximation Parsegian2006 . Within this approximation the
friction coefficient for a spherical tip with a radius $R$

$\Gamma_{s}=2\pi\int_{0}^{R}d\rho\rho\gamma(z(\rho))$ (28)

and for a cylindrical tip ith a radius $R$ and a width $w$

$\Gamma_{c}=2w\int_{0}^{R}d\rho\gamma(z(\rho))$ (29)

where $\gamma(z)$ is the friction coefficient between two plates separated by
a distance $z(\rho)=d+R-\sqrt{R^{2}-\rho^{2}}$ denoting the tip-surface
distance as a function of the distance $\rho$ from the tip symmetry. In the
extreme-near field the friction coefficient $\gamma\sim
10^{-12}-10^{-13}$kgs-1 which can be measured with the present state-of-the
art equipment. The blue and green lines include the contribution from the
electrical double layer, while the brown and pink lines without this
contribution, for a cylindrical and spherical tip, respectively. The
cylindrical tip has the radius $R=1\mu$m and the width $w=7\mu$m, the
spherical tip has the radius $R=30$nm. For $d<10$nm the contribution from the
electrical double layer produces the friction coefficient $\gamma\sim
10^{-10}-10^{-12}$kgs-1 that is by many orders of the magnitude larger than
the friction coefficient without this contribution. This friction can be
measured with the present state-of-the art equipment.

##  V Conclusion

We have shown that the interaction of the electromagnetic field with the
surface charge density of the electric double layer produces an enhancement of
the Casimir and electrostatic non-contact friction between a gold tip and a
gold plate by many orders of magnitude in comparison with the theory without
taking into account of this contribution. For electrostatic friction, which
are dominated by the excitation of Rayleigh surface waves, the theory is in
good agreement with experimental data. The theory also shows that the
electrical double layer contribution to the Casimir friction can be measured
using AFM.

The reported study was funded by RFBR according to the research project No
19-02-00453

<EMAIL_ADDRESS>
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