# Note: model title and parameter table are inserted automatically r""" Calculates the interparticle structure factor for a hard sphere fluid with a narrow, attractive, potential well. Unlike the :ref:`squarewell` model, here a perturbative solution of the Percus-Yevick closure relationship is used. The strength of the attractive well is described in terms of "stickiness" as defined below. The perturbation parameter (perturb), $\tau$, should be fixed between 0.01 and 0.1 and the "stickiness", $\epsilon$, allowed to vary to adjust the interaction strength. The "stickiness" is defined in the equation below and is a function of both the perturbation parameter and the interaction strength. $\epsilon$ and $\tau$ are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, the width of the square well, $\Delta$ (having the same units as $R$\ ), and the depth of the well, $U_o$, in units of $kT$. From the definition, it is clear that smaller $\epsilon$ means a stronger attraction. .. math:: \epsilon &= \frac{1}{12\tau} \exp(u_o / kT) \\ \tau &= \Delta / (\sigma + \Delta) where the interaction potential is .. math:: U(r) = \begin{cases} \infty & r < \sigma \\ -U_o & \sigma \leq r \leq \sigma + \Delta \\ 0 & r > \sigma + \Delta \end{cases} The Percus-Yevick (PY) closure is used for this calculation, and is an adequate closure for an attractive interparticle potential. The solution has been compared to Monte Carlo simulations for a square well fluid, with good agreement. The true particle volume fraction, $\phi$, is not equal to $h$ which appears in most of reference [1]. The two are related in equation (24). Reference [1] also describes the relationship between this perturbative solution and the original sticky hard sphere (or "adhesive sphere") model of Baxter [2]. .. note:: The calculation can go haywire for certain combinations of the input parameters, producing unphysical solutions. In this case errors are reported to the command window and $S(q)$ is set to -1 (so it will disappear on a log-log plot!). Use tight bounds to keep the parameters to values that you know are physical (test them), and keep nudging them until the optimization does not hit the constraints. .. note:: Earlier versions of SasView did not incorporate the so-called $\beta(q)$ ("beta") correction [3] for polydispersity and non-sphericity. This is only available in SasView versions 5.0 and higher. In SasView the effective radius may be calculated from the parameters used in the form factor $P(q)$ that this $S(q)$ is combined with. For 2D data the scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as .. math:: q = \sqrt{q_x^2 + q_y^2} References ---------- .. [#] S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 .. [#] R J Baxter, *J. Chem. Phys.*, 49 (1968), 2770-2774 .. [#] M Kotlarchyk and S-H Chen, *J. Chem. Phys.*, 79 (1983) 2461-2469 Authorship and Verification ---------------------------- * **Author:** * **Last Modified by:** * **Last Reviewed by:** Steve King **Date:** March 27, 2019 """ # TODO: refactor so that we pull in the old sansmodels.c_extensions import numpy as np from numpy import inf name = "stickyhardsphere" title = "'Sticky' hard sphere structure factor with Percus-Yevick closure" description = """\ [Sticky hard sphere structure factor, with Percus-Yevick closure] Interparticle structure factor S(Q) for a hard sphere fluid with a narrow attractive well. Fits are prone to deliver non- physical parameters; use with care and read the references in the model documentation.The "beta(q)" correction is available in versions 4.2.2 and higher. """ category = "structure-factor" structure_factor = True single = False # ["name", "units", default, [lower, upper], "type","description"], parameters = [ # [ "name", "units", default, [lower, upper], "type", # "description" ], ["radius_effective", "Ang", 50.0, [0, inf], "volume", "effective radius of hard sphere"], ["volfraction", "", 0.2, [0, 0.74], "", "volume fraction of hard spheres"], ["perturb", "", 0.05, [0.01, 0.1], "", "perturbation parameter, tau"], ["stickiness", "", 0.20, [-inf, inf], "", "stickiness, epsilon"], ] def random(): """Return a random parameter set for the model.""" pars = dict( scale=1, background=0, radius_effective=10**np.random.uniform(1, 4.7), volfraction=np.random.uniform(0.00001, 0.74), perturb=10**np.random.uniform(-2, -1), stickiness=np.random.uniform(0, 1), ) return pars # No volume normalization despite having a volume parameter # This should perhaps be volume normalized? form_volume = """ return 1.0; """ Iq = """ double onemineps,eta; double sig,aa,etam1,etam1sq,qa,qb,qc,radic; double lam,lam2,test,mu,alpha,beta; double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq; onemineps = 1.0-perturb; eta = volfraction/onemineps/onemineps/onemineps; sig = 2.0 * radius_effective; aa = sig/onemineps; etam1 = 1.0 - eta; etam1sq=etam1*etam1; //C //C SOLVE QUADRATIC FOR LAMBDA //C qa = eta/6.0; qb = stickiness + eta/etam1; qc = (1.0 + eta/2.0)/etam1sq; radic = qb*qb - 2.0*qa*qc; if(radic<0) { //if(x>0.01 && x<0.015) // Print "Lambda unphysical - both roots imaginary" //endif return(-1.0); } //C KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL radic = sqrt(radic); lam = (qb-radic)/qa; lam2 = (qb+radic)/qa; if(lam2test) { //if(x>0.01 && x<0.015) // Print "Lambda unphysical mu>test" //endif return(-1.0); } alpha = (1.0 + 2.0*eta - mu)/etam1sq; beta = (mu - 3.0*eta)/(2.0*etam1sq); //C //C CALCULATE THE STRUCTURE FACTOR //C kk = q*aa; k2 = kk*kk; k3 = kk*k2; SINCOS(kk,ds,dc); //ds = sin(kk); //dc = cos(kk); aq1 = ((ds - kk*dc)*alpha)/k3; aq2 = (beta*(1.0-dc))/k2; aq3 = (lam*ds)/(12.0*kk); aq = 1.0 + 12.0*eta*(aq1+aq2-aq3); // bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3); bq2 = beta*(1.0/kk - ds/k2); bq3 = (lam/12.0)*((1.0 - dc)/kk); bq = 12.0*eta*(bq1+bq2-bq3); // sq = 1.0/(aq*aq +bq*bq); return(sq); """ demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05, stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40) # tests = [ [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0, 'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1, 'radius_effective_pd': 0}, [0.001, 0.003], [1.09718, 1.087830]], ]