{"id":"c5f9dc60-ca5a-42d0-a759-d1b2968d40b4","arxiv_id":"2509.03342","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Depletion forces between large colloids are independent of large-particle density for size ratios below about 0.15, but show clear many-body effects at ratios 0.45 and 0.60.","lead":"This paper measures how depletion forces between large particles in a mixture are affected by other large particles nearby, using simulations and theory. It finds that for very small size ratios these many-body effects vanish, while for larger size ratios they are significant and can be quantified.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The mWCA potential at k_BT/epsilon=1 gives a small sphere order-unity Boltzmann weight in the triangular interstice of three large spheres at q=0.15, so Section II's geometric exclusion does not clearly extend; the claimed confirmation of the hard-sphere q3 threshold is not established.","rationale":"The paper's headline empirical result—phi_l independence of the depletion force at q=0.15 at fixed mu_s—is plausible and has internal support: the OZ-based calculation (Fig. 3(b)) quantitatively reproduces the CBF curves, and the q=0.45/0.60 cases (Figs. 4 and 5) demonstrate that the method does resolve phi_l dependence when it is present. I therefore see no basis to reject the empirical superposition itself, and the noMBC comparison in Fig. 8 is consistent with a small many-body contribution at q=0.15. The load-bearing weakness lies in the interpretation of the null as a confirmation of the hard-sphere geometric threshold. The mWCA potential at k_BT/epsilon=1 is measurably soft (the paper itself notes the finite slope of the repulsive branch and the reduced width of the attractive zone in Fig. 3(a)). For hard spheres, q3=0.1547 is exact because a small sphere cannot simultaneously touch three touching large spheres; for the mWCA potential at this state point, my estimate gives a three-body Boltzmann weight of order 0.5 for the compressed (0.95 sigma_l) triangle and 0.01 at 0.90 sigma_l, i.e., a channel that is suppressed but far from exponentially forbidden. Section II's 'extremely unlikely' premise therefore does not hold quantitatively, so the observed phi_l independence cannot be attributed to the hard-sphere mechanism without a direct check; it may reflect cancellation among soft-sphere many-body channels or limited statistical resolution. No error bars or quantitative bound on phi_l drift accompany Fig. 3(a), and the single q value is only 3% below the hard-sphere threshold. Separately, the Section I claim that the CBF scheme 'explicitly distinguishes' two-, three-, and higher-order contributions is not delivered by the presented MBC/noMBC contrast, which is a binary conditioned-average difference rather than the order-by-order decomposition outlined in Eq. (9). None of this overturns the paper—the empirical result may well be correct—but it justifies conditions: error bars on the null, a softness check (steeper potential or explicit three-body measurement), and a more measured statement of the separation claim. This sharpens the reader's weakest-assumption analysis, and the CONDITIONAL verdict stands.","tokens_in":21501,"tokens_out":30105,"duration_ms":262108,"concrete_test":"Simulate three large mWCA particles fixed in an equilateral triangle (side sigma_l, and also side 0.95 sigma_l) in a bath of small particles at the fixed mu_s values of Table I; average over small-particle configurations and measure the total force on one large particle. Subtract the bare mWCA forces from the two other large particles and the pair depletion forces taken from Fig. 3(a) at the corresponding distances. If the residual (explicit three-body force) at q=0.15 exceeds the statistical error of the pair forces, then the geometric exclusion of Section II fails for the soft potential and the phi_l-independence null cannot be claimed as confirmation of the hard-sphere q3 mechanism. A preliminary analytic check at beta*epsilon=1 gives a three-body Boltzmann weight exp(-0.71)~0.5 for the compressed (0.95 sigma_l) triangle, so the 'extremely unlikely' premise is already doubtful.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central null result—phi_l independence of the depletion force at q=0.15 under fixed mu_s—is presented as a confirmation of the hard-sphere geometric threshold q3=2/sqrt(3)-1=0.1547 (Section II). The bridge from hard spheres to the simulated mWCA particles is the statement that for semi-hard spheres 'the eventual touching of three larger spheres with a smaller one will be extremely unlikely.' At k_BT/epsilon=1 this premise is quantitatively doubtful. For three mWCA large spheres compressed to center distance r_ll=0.95 sigma_l (Boltzmann cost ~0.13 per large-large pair, ~0.39 total for the triangle), a small sphere of diameter 0.15 sigma_l placed at the triangle centroid sits at distance ~0.549 sigma_l from each large center, below the small-large contact distance sigma_sl=0.575 sigma_l; the small-large repulsion costs ~0.11 per contact, ~0.32 total, giving a combined three-body Boltzmann weight exp(-0.71)~0.5 relative to bulk. At r_ll=0.90 sigma_l the weight is still exp(-4.4)~0.01. The three-body configuration is thus populated with order-unity weight, not exponentially suppressed. Since Eq. (9) predicts a phi_l dependence through F^(3) rho_0 whenever such channels are open, the observed superposition in Fig. 3(a) cannot be attributed to the hard-sphere mechanism without a quantitative check. The figure shows no error bars and the text asserts only 'indistinguishable within the level of numerical error'; no bound is placed on a possible phi_l drift. Thus the load-bearing interpretation of the null as a confirmation of the geometric threshold is not established, even though the empirical superposition itself may be genuine.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies depletion forces in binary mixtures of quasi-hard (mWCA) spheres, with size ratios q=0.15, 0.45, and 0.60. The authors use the contraction-of-bare-forces (CBF) method in the semi-grand ensemble, fixing the chemical potential of the depletant by Widom insertion, and compare with Ornstein–Zernike integral-equation theory with a modified Verlet closure. The central claims are: (i) for q≤0.15 (below the hard-sphere geometric threshold q3=2/√3−1), the depletion force is independent of the large-particle packing fraction φ_l at fixed μ_s, confirming a purely geometric prediction; (ii) for q=0.45 and 0.60 the force depends strongly on φ_l and can be fitted/extrapolated with low-degree polynomials; and (iii) the CBF scheme distinguishes two-body from three- and higher-body contributions by comparing runs with and without triangular large-particle configurations. The paper also reports agreement between CBF and OZ theory.","tokens_in":21879,"tokens_out":7986,"duration_ms":70296,"significance":"The significance of a verified criterion for pair-additive depletion potentials is considerable for coarse-grained modeling of colloid–polymer mixtures. The paper's strengths are its direct force-based methodology (avoiding integration drift), the semi-grand canonical protocol, the systematic scan over φ_l and μ_s, and the independent OZ cross-check. The promise of an explicit order-by-order separation of many-body contributions, if realized, would be a substantial methodological advance. However, in its present form the evidence supports the qualitative trends but not the two strongest statements: the hard-sphere interpretation of the q=0.15 null result and the claim of an explicit two/three/higher-order separation. With the additional quantitative checks and claim revisions requested below, the paper would be a solid contribution to the soft-matter literature.","major_comments":[{"comment":"The paper's central confirmation claim rests on the assertion that the geometric threshold q3=2/√3−1 applies to the simulated mWCA particles at kBT/ε=1. Section II states that for semi-hard spheres \"the eventual touching of three larger spheres with a smaller one will be extremely unlikely,\" but no quantitative estimate is given. At q=0.15 the small–large contact distance is σ_sl=0.575σ_l, while three large spheres compressed to r_ll≈0.95σ_l create a triangular interstice whose centroid is about 0.549σ_l from each large center; the small–large repulsion at that point is already nonzero, so such three-body configurations are not exponentially suppressed at kBT/ε=1. To support the interpretation of Fig. 3(a) as a confirmation of the hard-sphere prediction, the authors need to characterize the effective softness of the mWCA potential (e.g., an effective hard-sphere diameter or Barker–Henderson mapping) and show that the threshold remains below q=0.15, or, alternatively, state the independence as an empirical observation independent of the geometric argument.","section":"Section II and IV.B"},{"comment":"The null result at q=0.15—that the depletion force is independent of φ_l under fixed μ_s—is asserted from visual superposition without error bars. The text says the curves \"become indistinguishable within the level of numerical error\" (Fig. 3 caption), but no measure of the numerical error or a bound on a possible φ_l drift is provided. The paper should show statistical error bars (the ten independent realizations mentioned in §IV.B make this straightforward) and, ideally, a quantitative comparison such as the maximum difference between φ_l curves divided by the estimated uncertainty. This is load-bearing because the abstract and conclusions claim verification \"with numerical precision.\"","section":"Fig. 3(a) and Section V.B"},{"comment":"The paper claims that the CBF scheme \"is able to explicitly distinguish between the two-, three-, and higher-order many-body contributions\" and calls this the most significant result. The actual analysis, however, compares the full force (including all many-body effects) with a force measured after excluding configurations in which large particles form triangles; it does not separately extract the f^(2), F^(3), F^(4), ... terms of Eq. (9). The comparison can at most separate \"with\" from \"without\" triangular configurations, and even that separation is not an order-by-order decomposition. Either provide a density-resolved fit of Eq. (9) that yields individual F^(n), or revise the claim to say that the method separates the total many-body contribution from the pair contribution.","section":"Section I (final paragraph), Section III, V.C"},{"comment":"The operational definition of the noMBC sampling is not given. Fig. 7 is a sketch; the text says \"configurations where large particles do not form triangles\" are discarded, but no threshold distance or graph criterion is specified for what counts as a triangle. Because the exclusion conditions the sample on the absence of triangular large-particle arrangements, it can bias the measured pair force (e.g., through correlations between pair distance and third-particle proximity). The authors should define the criterion precisely (e.g., three large particles with mutual separations below a cutoff r_c) and test the sensitivity of the noMBC results to r_c.","section":"Section V.C and Fig. 7"}],"minor_comments":[{"comment":"Throughout the text there are typos (\"a priory\" in Section I, \"interations\" in Ref. [8], \"Webwer\" in Ref. [2], \"unified\" in the abstract); the manuscript would benefit from a careful proofread.","section":"Throughout"},{"comment":"In Figs. 8 and 9 the legends use \"BMBC\" and \"WMBC\" but these abbreviations are never defined; the captions only define MBC and noMBC. These should be replaced by \"with MBC\" and \"without MBC\" or clarified.","section":"Figs. 8 and 9"},{"comment":"The caption of Table II states that error bars are \"not shown\" and can be large for C and D; since these coefficients are used for extrapolation in Fig. 6, the fitted coefficients should be reported with standard errors.","section":"Table II"},{"comment":"The discrepancy between the CBF and OZ noMBC results in Figs. 8(a) and 8(b) (no visible difference in CBF, clear reduction in OZ) is attributed to the inadequacy of removing the bridge function, but the text does not explain which of the two procedures is a more faithful estimate of the true noMBC force; a few sentences of quantitative discussion would help.","section":"Figs. 8(a) and 8(b)"},{"comment":"The figures use color gradients to represent φ_l values, which is difficult to read in print; additional line styles or symbols would improve clarity.","section":"Figures 3–5"},{"comment":"Reference [34] provides a virial-based mapping between hard-sphere and continuous potentials and could be used to justify or calibrate the mWCA potential as a hard-sphere surrogate; the authors only cite it, without applying it.","section":"Reference [34]"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth publishing after revision; I do not see grounds for rejection. The main risk is that the q=0.15 null result is overinterpreted as confirmation of the hard-sphere geometric threshold when the mWCA softness is not quantified. I would ask the authors to add a quantitative softness check and error bars; the conceptual overclaim about order-by-order separation should also be corrected. No concerns about novelty disclosure; citations to the authors' own prior work are appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real value is the systematic CBF/OZ comparison for depletion forces at q=0.45 and 0.60, plus a variant that removes triangular large-particle configurations to expose many-body effects. The polynomial fits and the extrapolation to infinite dilution are practical and useful. The OZ/mVerlet closure reproduces the CBF results well, which gives an independent consistency check. The chemical potential matching via Widom is careful, with multiple realizations and four-digit precision claimed.\n\nThe soft spots are real but varied. The central null result for q=0.15 is presented without error bars; 'indistinguishable within numerical error' is a visual judgment, and no bound is placed on a possible phi_l drift. That needs a quantitative statement. More importantly, the mWCA potential at kBT/epsilon=1 is measurably soft—the repulsive branch has a finite slope, and the stress-test calculation shows a small sphere sitting in the triangular interstice of three large spheres has a Boltzmann weight around 0.5 relative to bulk, not the 'extremely unlikely' claimed in Section II. So the independence may be genuine, but attributing it to the hard-sphere geometric threshold q3 is not supported. The geometric mechanism is not the only way to get the null, and the paper cannot distinguish them on current evidence.\n\nThe claim in Section I that the CBF scheme 'explicitly distinguish[es] between the two-, three-, and higher-order many-body contributions' is also stronger than what the MBC/noMBC comparison delivers. The noMBC procedure removes triangle configurations, but that is not an order-by-order decomposition in the sense of Eq. (9). The paper's own Section V.C acknowledges this limitation for the OZ route, but the abstract and intro do not.\n\nMinor but worth flagging: reference [54] has a placeholder page number (1234567), which needs fixing, and there are typos elsewhere. No code or data are provided, so the numerics are not independently checkable.\n\nOverall, the q=0.45 and q=0.60 results are solid and the OZ agreement gives confidence in the method. The load-bearing weakness is the interpretation of the q=0.15 null. I would send this to peer review, but the authors should be pushed to add error bars, quantify the effect of softness, and moderate the order-by-order claims.","headline":"Useful new data on density-dependent depletion forces for q=0.45 and 0.60, but the q=0.15 null result is overinterpreted as confirmation of the hard-sphere geometric threshold when the simulated particles are visibly soft.","tokens_in":22480,"tokens_out":3203,"would_cite":false,"duration_ms":30895,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Below a size ratio of about 0.15, the depletion attraction between large colloids is unaffected by how many large colloids are present, confirming a purely geometric prediction by direct simulation of quasi-hard-sphere mixtures.","keywords":["depletion forces","many-body contributions","effective pair potentials","binary colloid mixtures","semi-grand canonical ensemble","contraction of bare forces","size ratio threshold","integral equation theory"],"falsifier":"Repeat the fixed-$\\mu_s$ protocol at $q = 0.15$ with systematically steeper and softer repulsive potentials, or with an event-driven true-hard-sphere algorithm, and test whether the superposition of depletion-force curves across $\\phi_l = 0.1$ to $0.4$ survives in every case; if it breaks down at some softness, the geometric explanation of the null result fails even if the empirical rule remains useful. Independently, hold two large particles near contact with a third large particle present and measure the three-body depletion force directly: at $q$ just below $q_3$ it should vanish if the no-triple-contact argument is right.","tokens_in":21303,"feed_emoji":"🫧","tokens_out":20812,"duration_ms":155750,"temperature":0.7,"pith_summary":"The paper establishes when many-body effects can be ignored in depletion interactions, the entropic attraction between large colloidal particles immersed in a sea of much smaller ones. Using a force-contraction simulation scheme that can separate the contributions of pairs, triples, and larger groups, the authors show that for size ratios $q = \\sigma_s/\\sigma_l \\le 0.15$ the density of large particles has no effect on their mutual depletion force, provided the chemical potential of the small depletants is held fixed; the force curves for large-particle packing fractions $\\phi_l = 0.1$ to $0.4$ superpose within numerical error. This confirms a decades-old geometric argument: below $q_3 = 2/\\sqrt{3} - 1 \\approx 0.1547$ a small sphere can slip through the gap between three touching large spheres, so it can never couple three large particles at once. For less asymmetric mixtures ($q = 0.45$ and $0.60$) the large-particle density strongly modulates the force, with roughly a 20% spread at $q = 0.6$, and that density dependence is captured by low-degree polynomials. If correct, the work gives a directly testable criterion for when pair-additive depletion potentials are good enough, plus a method to quantify how much of any effective force comes from three or more particles.","feed_headline":"Below q=0.15, depletion forces ignore large-colloid crowding","feed_subtitle":"Simulations confirm the geometric threshold where large-colloid density stops changing their entropic attraction.","key_machinery":"The central mechanism is a geometric threshold for hard spheres: the size ratio $q_3 = 2/\\sqrt{3} - 1 \\approx 0.1547$ at which a small sphere just fits through the triangular cavity formed by three mutually touching large spheres; below it a depletant can never be in triple contact with large particles, so three-body and higher effective interactions are geometrically forbidden, and a second ratio $q_4 = \\sqrt{3/2} - 1 \\approx 0.2247$ marks where four-body terms switch on. The simulation engine is the contraction of bare forces (CBF), a least-squares inversion that extracts effective pair forces $G(r)$ from the noisy instantaneous forces on large particles averaged over configurations of the small ones; by resampling only configurations free of large-particle triangles, the same engine produces a many-body-free version of the force. The semi-grand ensemble is the enabling thermodynamic frame: the small-particle chemical potential $\\mu_s$ is measured by a particle-insertion method in a reservoir and held constant while $\\phi_l$ varies, with the small-particle packing fraction $\\phi_s$ adjusted accordingly, which is what makes the predicted independence from $\\phi_l$ a well-posed, testable statement. The Ornstein-Zernike integral-equation approach with the modified-Verlet closure supplies an independent, much cheaper route to the same effective forces.","core_discovery":"On the paper's own terms, the central discovery is that the many-body content of a depletion force is a quantitative, size-ratio-controlled quantity, and that it can be measured order by order. In the semi-grand ensemble, with the chemical potential of the small particles fixed by a particle-insertion method, the CBF simulations show that the effective two-body depletion force between large colloids is independent of the large-particle packing fraction $\\phi_l$ whenever $q \\le q_3 = 2/\\sqrt{3} - 1 \\approx 0.1547$: the curves for $\\phi_l = 0.1, 0.2, 0.3, 0.4$ superpose within numerical error, directly verifying the geometric prediction that a small particle can never simultaneously touch three large ones below this threshold. The same scheme, by discarding configurations in which large particles form triangles, isolates the purely two-body contribution and shows three-body terms are negligible at $q = 0.15$ yet responsible for the density effect at $q = 0.60$. For $q = 0.45$ and $0.60$ the depletion force weakens as $\\phi_l$ grows, about a 20% spread between $\\phi_l = 0.1$ and $0.4$ at $q = 0.6$, and this dependence is well described by third-order polynomials in $\\phi_l$ whose coefficients can extrapolate the force to the infinite-dilution limit $\\phi_l \\to 0$. The independent Ornstein-Zernike integral-equation route, closed with the modified-Verlet approximation, quantitatively reproduces the CBF results, and the authors present the explicit separation of two-, three-, and higher-order contributions as the work's most significant outcome.","pith_inferences":["Applying the same fixed-$\\mu_s$ protocol at $q \\approx 0.2$, between $q_3$ and $q_4$, should reveal a staircase of many-body orders: two- and three-body terms present, four-body terms still absent. The paper does not run this test.","The superposition test itself could serve as a calibrated probe of how faithfully a soft potential mimics hard-sphere geometry, since softness visibly rounds the force curves and may shift the effective threshold.","The screening of the depletion attraction by large-particle density suggests a density-dependent effective pair potential, which coarse-grained dynamical simulations could absorb as a local rescaling of the attraction.","The close agreement between the integral-equation route and CBF in the many-body regime suggests the cheap theory could screen parameter space for simulations, while the residual discrepancies when bridge functions are removed locate precisely where the closure relation needs improvement."],"forward_implications":["For any binary mixture with $q \\le 0.15$, effective potentials are pair-additive up to large-particle packing fractions of at least 0.4, so models of phase behavior in that regime can omit three-body and higher terms.","At fixed $\\mu_s$, a depletion-force measurement made once at low large-particle density transfers unchanged to large-particle concentrations across the investigated range (up to $\\phi_l = 0.4$), shortening the simulation or experimental effort needed to characterize such mixtures.","The published third-order polynomial coefficients for $q = 0.45$ and $0.60$ let other researchers interpolate depletion forces at un-simulated packing fractions and extrapolate them to $\\phi_l \\to 0$ without running new simulations.","Because the effective two-body force is shown to carry many-body contributions at the bare-interaction level, the CBF decomposition yields an explicit budget of how much of any measured attraction is pairwise versus three-body and higher at each state point.","For $q > q_3$, the weakening of the attraction as dense aggregates form could frustrate depletion-driven phase transitions, with condensation weakening the very attraction that drives it, a possibility the authors flag as worth exploring."],"supporting_citations":[{"why":"Defines the contraction-of-bare-forces (CBF) scheme that this paper extends to separate two-, three-, and higher-order many-body contributions.","marker":"[14]"},{"why":"Prior CBF study of depletion forces beyond the dilute limit; supplies the baseline method and framing this work builds on.","marker":"[15]"},{"why":"Early statement that effective interactions in highly asymmetric hard-sphere mixtures are almost free of many-body effects.","marker":"[31]"},{"why":"Source of the geometric $q_3$ threshold argument and the claim that pair interactions nearly exhaust the effective potential for $q < q_3$.","marker":"[32]"},{"why":"Establishes that in the semi-grand ensemble the depletion force becomes a function of the depletant chemical potential only for $q < q_3$.","marker":"[33]"},{"why":"Integral-equation theory for effective interactions, the independent theoretical route whose predictions are compared with the simulations throughout.","marker":"[25]"},{"why":"Modified-Verlet closure relation used to solve the Ornstein-Zernike equations; the accuracy of the theoretical depletion forces rests on it.","marker":"[27]"},{"why":"Second-virial mapping used to justify the modified Weeks-Chandler-Andersen potential as a faithful stand-in for hard spheres.","marker":"[34]"},{"why":"Particle-insertion technique for measuring chemical potentials, the method the paper uses to fix the depletant chemical potential at its reservoir value.","marker":"[44]"}],"fun_headline_variants":["Depletion forces become pairwise-only below q=0.155","Many-body depletion terms turn on only above q=0.155","Simulations quantify pair, triple, and higher depletion forces","Below q=0.155, big-colloid crowding does not affect depletion","New method isolates three-body depletion forces from simulations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise, stated in Section II as 'the previous geometrical arguments extend approximately to semi-hard spheres,' is that the slightly soft repulsive potential used in the simulations behaves enough like hard spheres for the geometric threshold $q_3 = 2/\\sqrt{3} - 1 \\approx 0.1547$, derived for perfect hard spheres, to apply unchanged to the simulated particles; the finite slope of the repulsive force in the measured curves shows the softness is real, and the paper provides no systematic check of how it shifts the effective threshold.","fun_headline_variants_meta":{"raw":{"variants":["Depletion forces become pairwise-only below q=0.155","Many-body depletion terms turn on only above q=0.155","Simulations quantify pair, triple, and higher depletion forces","Below q=0.155, big-colloid crowding does not affect depletion","New method isolates three-body depletion forces from simulations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000937,"raw_usage":{"total_tokens":4127,"prompt_tokens":1186,"completion_tokens":2941,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":802,"completion_tokens_details":{"reasoning_tokens":2855}},"tokens_in":802,"tokens_out":2941,"duration_ms":20344,"temperature":1.0,"reasoning_tokens":2855,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:30:22.991947+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the fixed-$\\mu_s$ protocol at $q = 0.15$ with systematically steeper and softer repulsive potentials, or with an event-driven true-hard-sphere algorithm, and test whether the superposition of depletion-force curves across $\\phi_l = 0.1$ to $0.4$ survives in every case; if it breaks down at some softness, the geometric explanation of the null result fails even if the empirical rule remains useful. Independently, hold two large particles near contact with a third large particle present and measure the three-body depletion force directly: at $q$ just below $q_3$ it should vanish if the no-triple-contact argument is right.","supporting_citations":[{"cited_title":"Santos, M","cited_arxiv_id":null,"evidence_quote":"Defines the contraction-of-bare-forces (CBF) scheme that this paper extends to separate two-, three-, and higher-order many-body contributions."},{"cited_title":"Dobnikar, R","cited_arxiv_id":null,"evidence_quote":"Prior CBF study of depletion forces beyond the dilute limit; supplies the baseline method and framing this work builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Early statement that effective interactions in highly asymmetric hard-sphere mixtures are almost free of many-body effects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the geometric $q_3$ threshold argument and the claim that pair interactions nearly exhaust the effective potential for $q < q_3$."},{"cited_title":"Dijkstra, R","cited_arxiv_id":null,"evidence_quote":"Establishes that in the semi-grand ensemble the depletion force becomes a function of the depletant chemical potential only for $q < q_3$."},{"cited_title":"Dijkstra, R","cited_arxiv_id":null,"evidence_quote":"Integral-equation theory for effective interactions, the independent theoretical route whose predictions are compared with the simulations throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Modified-Verlet closure relation used to solve the Ornstein-Zernike equations; the accuracy of the theoretical depletion forces rests on it."},{"cited_title":"Dijkstra, R","cited_arxiv_id":null,"evidence_quote":"Second-virial mapping used to justify the modified Weeks-Chandler-Andersen potential as a faithful stand-in for hard spheres."},{"cited_title":"Eslami and F","cited_arxiv_id":null,"evidence_quote":"Particle-insertion technique for measuring chemical potentials, the method the paper uses to fix the depletant chemical potential at its reservoir value."}],"review_version":2}