Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions

Alessio Squarcini; Anna Macio{\l}ek; Erich Eisenriegler; Siegfried Dietrich

REVIEW 2 major objections 5 minor 72 references

Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives exact order-parameter, energy-density, and stress-tensor profiles around Janus-type particles in a critical two-dimensional Ising fluid and uses them to obtain asymptotically exact, orientation-dependent critical Casimir…

desk verdict Exact 2D CFT profiles and SPOE asymptotics for Janus/quadrupole colloids are credible and worth refereeing; the SPOE completeness caveat is real but not fatal. read the letter →

arxiv 1908.06950 v1 pith:5ZN4WBDI submitted 2019-08-12 cond-mat.stat-mech cond-mat.soft

Alessio Squarcini , Anna Macio{l}ek , Erich Eisenriegler , Siegfried Dietrich This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.soft

keywords criticalCasimireffectJanusparticlespatchycolloidsconformalfieldtheoryIsingmodelbinaryliquidmixturesmallparticleoperatorexpansiontorque

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

At the consolute point of a binary liquid mixture, the solvent is a critical Ising fluid, and in two dimensions conformal invariance is powerful enough to fix the exact effect of a colloidal inclusion. This paper derives closed-form, universal profiles of the order parameter, energy density, and stress tensor around colloids whose surfaces carry alternating + and - adsorption patches, including symmetric and generalized Janus circles, Janus needles, and circular quadrupoles. It then packages the single-particle effect into a small particle operator expansion, a multipole series of local operators at the particle's center, which yields asymptotically exact expressions for the critical Casimir free energy of a small particle near walls, strips, wedges, or another particle, including the full dependence on orientation. The orientation dependence implies Casimir torques and preferred alignments, and the analytical two-particle results match recent lattice simulations at the qualitative level.

What carries the argument

The central object is the conformal transformation from the exterior of the particle to the upper half plane; for a unit circle it is $w(z)=i(z+1)/(z-1)$, with the particle's + and - arcs becoming + and - half-lines on the real axis. The paper combines exact half-plane one- and two-point functions for such alternating boundary patterns with the conformal transformation laws for primary operators, involving the factor $|dw/dz|^{x_O}$, and for the stress tensor, which includes the Schwarzian derivative. The small particle operator expansion is the second engine: it encodes the particle's distant effect as $1+s_P$, with $s_P$ a series of descendants of the energy, order-parameter, and identity operators located at the particle center, and its prefactors are fixed by matching the exact isolated-particle profiles and selected two-point functions. This reduces confinement and pair interactions to a handful of bulk or confined correlation-function averages and produces the explicit orientation dependence through the spin of each descendant.

What would settle it

An exact transfer-matrix calculation of the critical Casimir free energy of a Janus circle in an annulus with an inhomogeneous boundary would settle the main claim: the expansion predicted by the small particle operator expansion must match the exact large-distance series in the size-to-distance ratio through the quoted orders. A mismatch in the leading orientation-dependent term would show that the undetermined, critical-point-vanishing operators are not subleading.

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Extended reading notes

Core claim

For a patchy particle in the plane at the bulk critical point, the paper establishes that every one-point function outside the particle is obtained exactly by mapping the exterior to the upper half plane and reading off the known half-plane profile for the corresponding alternating boundary pattern. For a unit Janus circle this gives the order-parameter profile in Eq. (2.15), the energy profile in Eq. (2.16), and the stress profile in Eq. (2.18); analogous exact results hold for generalized Janus circles, quadrupoles, and Janus needles. The same conformal data fix every prefactor in the small particle operator expansion, so that the free energy of transfer into a half plane, strip, or wedge, and the interaction between two distant particles, are exact through the quoted orders in the relevant small size-to-distance ratios, with the orientation angle entering through the spins of the descendant operators. Consequences include non-monotonic insertion free energies, orientation-selecting torques, and a stitch-together Derjaguin rule for near-contact Janus circles.

Load-bearing premise

The load-bearing premise is that the small particle operator expansion, with its prefactors fixed at the critical point, is complete enough that the undetermined off-critical operator terms never enter at the order of the leading or next-to-leading interactions kept in the paper; if they do, the predicted free energies and preferred orientations could change.

Editorial extensions

If this is right

A Janus particle in an ordinary half-plane orients so that its switching points face the wall, while in an ordinary strip its dipole aligns parallel to the strip axis.
For a Janus circle facing a + wall, the insertion free energy is repulsive at large distances and attractive at short distances, implying a maximum in between; for an ordinary wall the opposite trend implies a minimum and a stable standoff distance.
Two Janus circles attract or repel depending on relative orientation, and near contact the leading Derjaguin force is the arithmetic mean of the forces for the two homogeneous boundary pairs, with a next-to-leading correction of order $(C/R)^{1/2}$.
A quadrupolar particle in a + wedge switches its preferred orientation at a wedge opening angle of about $123.5^\circ$, giving a concrete geometry-controlled orientational transition.
The same expansion produces explicit higher-body contributions, such as the subleading three-body term between two Janus circles and an ordinary circle, showing that critical Casimir many-body forces are accessible in this framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the explicit orientation dependence of the free energy directly yields a Casimir torque, the negative derivative of the free energy with respect to the orientation angle; extracting this torque as a function of distance and boundary type gives a concrete prediction for orientational ordering of Janus colloids.
Since conformal invariance is special to two dimensions, the quantitative profiles will not transfer to three-dimensional Janus spheres, but the small particle operator expansion logic suggests that analogous expansions built from exact single-sphere profiles could supply the leading orientation-dependent Casimir terms in $d=3$.
The decreasing energy-prefactor across Janus, quadrupolar, and ordinary circular boundaries hints that particles with many alternating patches cross over to effectively free-boundary behavior; computing profiles for particles with more switches would test this crossover directly.
A lattice-simulation test in the Janus-needle geometry, which suffers fewer shape-discretization artifacts than circular particles on a square lattice, would cleanly isolate the predicted leading $R^{9/8}$ orientation-dependent term near a symmetry-breaking wall.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper gives exact one-point profiles for the order parameter, energy density, and stress tensor around Janus circles, Janus needles, generalized Janus particles, and quadrupoles in the 2D critical Ising model, then builds a small particle operator expansion (SPOE) to get asymptotically exact position- and orientation-dependent critical Casimir free energies in confining geometries and between two particles. The single-particle profiles are the real prize: they follow from known half-plane correlators via explicit conformal maps, with no fitted parameters, and the consistency checks in the appendices are meaningful. The SPOE machinery is standard and carefully executed; matching the prefactors against one- and two-point functions is a legitimate way to fix them, and the resulting interaction free energies are the first analytic benchmarks for orientation-dependent Casimir forces between patchy colloids in 2D. This deserves a serious referee.

The soft spots are real but manageable. The SPOE completeness is asserted rather than proved: footnote 3 concedes that some operator prefactors cannot be determined, and the paper assumes those contribute only at higher order. The stress-test worry about a missed same-order operator would be serious if it landed, but it does not quite land. The SPOE is required to reproduce all correlation functions in the presence of the particle, and the operators included respect the particle symmetries; a critical operator of the same dimension that is compatible with the symmetries would show up in the exact profiles or in the two-point functions used for disentangling. The real gap is that the authors do not explicitly rule out such an operator. This is a limitation to state, not a reason to doubt the central results. The comparison with simulation is qualitative only, and the paper extrapolates asymptotic expressions to distances where the small-particle expansion is not controlled; the authors are honest about this, but readers should not treat the close-distance curves as quantitative. Appendix E also leans on unpublished material, which should be resolved before publication.

I would send this to peer review. The exact profiles alone justify it, and the SPOE results are likely to be widely used as benchmarks. I would ask the referees to require a clearer statement of the completeness assumption, a check that no same-dimension symmetry-compatible operator is missing, and either a proof or a citation for the unpublished Appendix E results. None of these are fatal; they are the normal conditions for a careful asymptotic construction.

Referee Report

2 major / 5 minor

Summary. The paper studies colloidal particles with chemically inhomogeneous surfaces suspended in a two-dimensional critical Ising fluid at its bulk critical point. Using conformal maps from the upper half plane with alternating +/− boundary conditions, it derives exact single-particle profiles of the order parameter, the energy density, and the stress tensor for symmetric and generalized Janus circles, Janus needles, and quadrupolar particles (Sec. 2). It then constructs a small-particle operator expansion (SPOE) whose prefactors are fixed by matching isolated-particle one- and two-point functions (Sec. 3 and Appendices B–D), and uses this SPOE to compute insertion free energies in half planes, strips, and wedges (Secs. 5–6) as well as pair interactions between distant particles (Sec. 7). The paper also compares the large-distance results with Derjaguin-type short-distance expressions and with simulation data from Ref. [57].

Significance. If the SPOE completeness and transferability assumptions hold, the paper delivers parameter-free universal predictions for position- and orientation-dependent critical Casimir interactions, including torques, that go beyond earlier homogeneous-particle results. The single-particle profiles in Sec. 2 are exact at all distances outside the particle and constitute a useful resource in their own right. The disentanglement of the third-order order-parameter descendants in Appendix B and the reproduction of the quadrupole stress-tensor profile from the identity-descendant sector in Appendix D are nontrivial consistency checks. The main open risk is that the interaction free energies are only as reliable as the truncated SPOE, whose completeness is asserted rather than proved; this is the load-bearing premise for the central claims of asymptotic exactness.

major comments (2)

[Sec. 3, Eq. (3.18); footnote 3] The interaction free energies in Secs. 5–7 are described as asymptotically exact, but the only justification for the truncated SPOE is the matching condition in Eq. (3.18). The prefactors are fixed so that the expansion reproduces the isolated-particle one-point functions and the selected two-point functions treated in Appendices B and D; operators that do not enter these correlators are left unconstrained. Footnote 3 explicitly concedes the existence of operator contributions with prefactors that vanish at the critical point and 'cannot be determined as described,' with the assumption that they are higher order. A missed operator of this kind, or a higher-spin identity descendant omitted from Eqs. (3.2)–(3.13), could acquire a nonvanishing average in a strip, wedge, or two-particle geometry at the same order as the retained R^3, R^{25/8}, or R^4 terms, thereby changing the predicted equilibrium orientations in Eqs. (5.26), (5.31), (6.8), and the pair potentials in Eq. (7.3). Since this is the step that carries the claim of asymptotic exactness, the manuscript should either provide a concrete argument (or an independent check, e.g., against an exact free-fermion solution in a special confined geometry) that all omitted operators are strictly higher order in every geometry considered, or explicitly downgrade the claims to leading-order asymptotics with an estimate of the omitted contributions.
[Sec. 4, Eqs. (4.1)–(4.3)] The transition from the solitary particle to a confined or two-particle geometry assumes that the SPOE operator series, with prefactors fixed from bulk-fitted isolated-particle data, persists unchanged in the presence of distant boundaries or a second particle. This is plausible for a local effective Hamiltonian, but it is an assumption that is not tested by the exact profiles of Sec. 2, which are the paper's strongest results. The manuscript contains no check of the SPOE in a geometry where independent results are available, even though exact strip data for homogeneous inclusions are available from Eqs. (5.21)–(5.22) and the literature cited there. Please add such a check, or state explicitly that this transferability is part of the SPOE ansatz and that the interaction results inherit this assumption; as written, the phrase 'asymptotically exact' in the abstract overstates what has been established.

minor comments (5)

[Abstract] The phrase 'in two spatial dimension' should be 'in two spatial dimensions.'
[Sec. 5.2.2, text after Eq. (5.25)] The sentence reporting the orientation for a ++ strip says 'α = 0 for Y0 < 0 and α = π for Y0 < 0'; the second inequality should presumably be Y0 > 0, since otherwise no orientation is predicted for Y0 > 0.
[Eq. (5.8) and surrounding text] The notation D ≡ D/(2x0) reuses the symbol D for both the needle length and a dimensionless ratio; this is confusing and should be replaced by a distinct symbol such as \tilde D.
[Eq. (2.49)] The symbol 'cc' is used to denote the complex conjugate without being defined in the text or in the glossary of Appendix A; please define it explicitly.
[Appendix B, Eq. (B.10)] The derivation of the prefactors A, B, and C is summarized very briefly; since this is the key step that disentangles the third-order descendants, please state explicitly that the expansion of Eq. (B.5) was matched to Eq. (B.3) (for example, by symbolic computation) so that the result is auditable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step: exact profiles come from external conformal inputs, SPOE coefficients are fitted to those profiles, and the wall, strip, wedge, and two-particle free energies are genuinely different geometric averages; footnote 3 is a stated completeness caveat rather than a circular reduction.

full rationale

The claimed exact single-particle profiles are not circular. They are obtained by applying the conformal transformation formula in Eq. (2.1), with the M\"obius maps in Eqs. (2.6) and (2.35), to half-plane one-point functions taken from Burkhardt, Guim, and Xue (Refs. [42], [55], [56]); the target interaction free energies never enter this construction. The SPOE coefficients are indeed fixed by requiring Eq. (3.18) to reproduce the isolated-particle profiles and selected two-point functions, as stated in Sec. 3 and Appendices B-D. That is a coefficient fit for an operator expansion, not a prediction of the quantities being matched. The subsequent interaction free energies are obtained from wall, strip, wedge, or two-particle averages of those operators via Eqs. (4.2), (4.3), and (7.2), which involve conformal data different from the fitted isolated-particle correlators; the distance and orientation dependences are therefore new content, not the fit themselves. Footnote 3 explicitly concedes that additional operators with prefactors vanishing at the critical point cannot be determined and are expected to give only higher-order corrections. This is an unproven completeness assumption that limits the force of the phrase 'asymptotically exact', but it is a correctness or rigor caveat, not an equation-level circular reduction: no displayed equation makes a target interaction equal by construction to an input used to fix the SPOE. The paper also cites earlier SPOE work by the same authors, but the expansion is re-validated internally against exact two-point functions and against independent simulation data, so the self-citations are not load-bearing unverified assumptions. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The calculation imports established CFT technology: conformal mapping, known half-plane correlators, and standard Virasoro algebra. The paper adds no free parameters and no invented entities; its own assumptions are the validity and completeness of the SPOE for inhomogeneously patterned surfaces.

assumptions (5)

domain assumption The 2D critical binary mixture belongs to the Ising universality class and is conformally invariant at the bulk critical point.
Invoked throughout Sec. 2 via the transformation formula Eq. (2.1), and required for all exact profile and correlation function results.
domain assumption Exact correlation functions for the upper half plane with alternating +/− boundary segments (Burkhardt-Guin-Xue, Refs [42,55,56]) are correct and are used as input without re-derivation.
The particle profiles in Secs. 2.1-2.4 are obtained by conformal transformation of the half-plane results in Eqs. (2.13), (2.14), (2.44), (2.45), and (2.53); any error in those inputs propagates to all downstream results.
domain assumption Surface patches act as infinitely strong + or − surface fields (normal boundary conditions), and free boundaries act as ordinary (O) boundary conditions.
The entire classification of boundary conditions and the universal amplitudes in Eq. (2.5) and (1.2) rely on this surface universality class picture.
ad hoc to paper The small-particle operator expansion provides a complete basis of conformal operators at the particle position, and prefactors determined from the isolated particle persist in the presence of distant boundaries.
The SPOE construction in Sec. 3, Eq. (3.18), and its use in Sec. 4 require this completeness and transferability; the paper itself flags in footnote 3 that some operator prefactors cannot be determined by its method.
standard math Level-2 degeneracy of primary operators in minimal models (Eq. (3.17), Appendix F) holds.
Used to eliminate L-2 type descendants in favor of ∂^2 operators in the SPOEs, Eqs. (3.2)-(3.13), and to simplify several correlation function computations.

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Cite this review

Pith. "Pith review of Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions." pith.science (2026). https://pith.science/paper/5ZN4WBDI

@misc{pith2026190806950,
  author       = {Pith},
  title        = {Pith review of: Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZN4WBDI}},
  note         = {Machine review of arXiv:1908.06950}
}

read the original abstract

We study colloidal particles with chemically inhomogeneous surfaces suspended in a critical binary liquid mixture. The inhomogeneous particle surface is composed of patches with alternating adsorption preferences for the two components of the binary solvent. By describing the binary liquid mixture \emph{at} its consolute point in terms of the critical Ising model we exploit its conformal invariance in two spatial dimension. This allows us to determine exactly the universal profiles of the order parameter, the energy density, and the stress tensor as well as some of their correlation functions around a single particle for various shapes and configurations of the surface patches. The formalism encompasses several interesting configurations, including Janus particles of circular and needle shapes with dipolar symmetry and a circular particle with quadrupolar symmetry. From these single-particle properties we construct the so-called small particle operator expansion (SPOE), which enables us to obtain asymptotically exact expressions for the position- and orientation-dependent critical Casimir interactions of the particles with distant objects, such as another particle or the confining walls of a half plane, strip, or wedge, with various boundary conditions for the order parameter. In several cases we compare the interactions at large distances with the ones at close distance (but still large on the molecular scale). We also compare our analytical results for two Janus particles with recent simulation data.

Figures

Figures reproduced from arXiv: 1908.06950 by the authors.

**Figure 1.** Circular and rod-like colloids with various patterns of chemical inhomogeneity. A symmetric Janus particle (a) and its generalized version (b) specified by the angle χ. (c) A “double” Janus particle, corresponding to a quadrupole. (d) A needle-shaped Janus colloid. α W+ θ r z0 0 2R (a) α Sab a b W y 2R (b) 2R α1 α2 r (c) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Typical configurations of Janus and quadrupolar pa [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The domain outside the circular particle centered at the origin in the z-plane and its boundary depicted in (b) are conformally mapped by the transformation w(z) in Eq. (2.6) to the upper half w-plane and its boundary depicted in (a). In particular, the switching point z = −1 on the particle boundary is mapped to its counterpart at the origin of the w-plane; a green dot marks its location. The circles shown in (b) a… view at source ↗

Figures from the paper (23 more)

**Figure 4.** Figure 4: Order parameter profiles for various realizations of the generalized Janus particle Jχ in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Energy density profiles for various realizations of the generalized Janus particle Jχ in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Stress tensor profile around the circular, symmetric Janus particle. (a) The normalized eigenvectors eb+(x, y) and eb−(x, y) of the stress tensor corresponding to the positive and negative eigenvalue, in red and blue, respectively. (b) The normalized eigenvector eb+(x…

**Figure 7.** Figure 7: Stress tensor profile around a generalized circular Janus particle. The arrangement and the meaning of the panels are the same as in [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Order parameter profile hΦ(z, z¯)iQ around the quadrupole particle (b) and the corresponding profile hΦ(w, w¯)iH−+−+ in the upper half w-plane (a). The order parameter vanishes along the dotted blue lines and diverges upon approaching the real axis in (a) and the parti…

**Figure 9.** Figure 9: Energy density profile hε(z, z¯)iQ around a quadrupole particle (b) and the corresponding profile hε(w, w¯)iH−+−+ in the upper half w-plane (a). The energy density vanishes along the dotted blue lines and diverges towards +∞ at the switching points, as indicated by the…

**Figure 10.** Figure 10: Stress tensor profile for the quadrupolar particle Q. The arrangement and meaning of the panels are like those in Figs. 6 and 7. 21 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 1.** Figure 1: α α α [PITH_FULL_IMAGE:figures/full_fig_p024_1.png]

**Figure 11.** Figure 11: Illustration of the rotation angle α. The meaning of the operators L−3Φ and L¯−3Φ in Eqs. (3.3) and (3.6), respectively, and of 23 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: Distance dependence of the insertion free energy δF of a Janus circle (a) and a Janus needle (b) with fixed orientation α = π/2 so that it faces with its + segment an ordinary boundary wall of the embedding right half plane. Combining the (extrapolated) large distance…

**Figure 13.** Figure 13: Same as [PITH_FULL_IMAGE:figures/full_fig_p032_13.png]

**Figure 14.** Figure 14: Same as [PITH_FULL_IMAGE:figures/full_fig_p033_14.png]

**Figure 15.** Figure 15: A wedge with π/Ξ = 2π/3 (a) and a wedge with π/Ξ = 3π/2 (b) in the z plane. The latter corresponds to an acute opening angle of π/2 at the apex in the inaccessible region (white). We show the curved rectangular grid which, via Eq. (5.3), corresponds to the grid of str…

**Figure 16.** Figure 16: Two identical Janus particles with radii [PITH_FULL_IMAGE:figures/full_fig_p042_16.png]

**Figure 17.** Figure 17: Five special angular configurations of two Janus p [PITH_FULL_IMAGE:figures/full_fig_p043_17.png]

**Figure 18.** Figure 18: The free energy of interaction (per kBTc) for two circular Janus colloids for the five orientations shown in [PITH_FULL_IMAGE:figures/full_fig_p044_18.png]

**Figure 19.** Figure 19: Two identical Janus needles with length D and center-to-center distance r. A nonzero needle thickness has been introduced for reasons of clarity. The orientations αj with j = 1, 2 are measured relative to the reference directions, as described in [PITH_FULL_IMAGE:fig…

**Figure 20.** Figure 20: Large distance behavior of the interaction energ [PITH_FULL_IMAGE:figures/full_fig_p046_20.png]

**Figure 21.** Figure 21: Configuration of an ordinary needle N of length [PITH_FULL_IMAGE:figures/full_fig_p046_21.png]

**Figure 22.** Figure 22: Large distance behavior of the free energy of inte [PITH_FULL_IMAGE:figures/full_fig_p047_22.png]

**Figure 23.** Figure 23: Two identical quadrupolar particles with radii [PITH_FULL_IMAGE:figures/full_fig_p048_23.png]

**Figure 24.** Figure 24: Four special angular configurations of quadrupol [PITH_FULL_IMAGE:figures/full_fig_p049_24.png]

**Figure 25.** Figure 25: The free energy of interaction (per kBTc) for two quadrupoles with the special orientations shown in [PITH_FULL_IMAGE:figures/full_fig_p049_25.png]

Discussion (0). Continue with ORCID to comment.

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Burkhardt T W and Eisenriegler E unpublished material 57 Appendix A Glossary and list of symbols In order to make the reading of the manuscript easier, here we summarize the mathematical symbols and abbreviations used throughout the main text of t his work. Certain extra symbo...
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(5.3), the wedge average⟨T (z)⟩Waa = ( 1− Ξ2) /(48z2) of T , which follows from Eq

Here we use the transformation in Eq. (5.3), the wedge average⟨T (z)⟩Waa = ( 1− Ξ2) /(48z2) of T , which follows from Eq. (5.27), and the averages in Eqs. (D.22) and (D.24) of T ¯T, L2 −2I, and L−4I. Further conﬁrmation of the expression in Eq. (D.25) stems fr om the bulk thre...
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actually is exact right at the critical point. Here we extend this reasoning to close circles with inhomogeneous boundary conditions, such as the circular Janus particles c onsidered here, realizing that the stress tensor in the corresponding strip can be calculated rather eas...
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ﬁt together

In this case Eqs. (E.3) and (E.2) imply C R = ñÅ R> R< ã1/ 4 − Å R< R> ã1/ 4ô2 , 2 R = 1 R< − 1 R> . (E.4) (ii) For R> = R0, circle 2 reduces to the imaginary axis, forming the boundar y of the right half z plane, which contains circle 1 with radius R1≡ R and its center at z =...

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